Magma's arithmetic geometry machinery spans dozens of Handbook chapters. Rather than diving deep into any one of them (there is a dedicated Elliptic Curves lesson for that), this card is a map: what each family of chapters is for, how the pieces relate, and which type or intrinsic to reach for when a problem shows up. Roughly, the territory splits into "curves and their arithmetic invariants" and "modular objects that explain or compute those invariants."
Curves of low genus: conics, genus one, and plane quartics
At genus zero, CrvCon (conics) and CrvRat (rational curves) are the special types. The central question is whether a genus zero curve has a rational point; if so it can be parametrised by a projective line, and if not, the obstruction is measured by bad primes and an associated quaternion algebra. Genus one curves get their own generalization: a ModelG1 of degree 2, 3, 4, or 5 represents a principal homogeneous space for an elliptic curve in the Weil-Chatelet group, embedded in projective space by a linear system — this is the machinery behind descent calculations rather than a curve type itself. At the other end of the genus scale, CrvG3 handles plane quartics (genus 3 non-hyperelliptic curves) through the Dixmier-Ohno invariants, a set of generators for the ring of invariants of ternary quartic forms that classify quartics up to isomorphism, much as j-invariants classify elliptic curves.
Elliptic curves across different base fields
The core elliptic curve chapter covers curves over general fields: Weierstrass models, group law arithmetic, and the shared invariants (conductor, local data). Three companion chapters specialise this to a particular base: finite fields (point counting via SEA or canonical lifts, built with elliptic curve cryptography in mind), the rationals and number fields (minimal models, Tate's algorithm, Mordell-Weil computation via descent, and analytic tools like modular parametrisations and Heegner points), and function fields (local invariants, minimal models, and Mordell-Weil for curves over rational function fields over finite fields). A typical invariant lookup looks the same regardless of which specialised chapter backs it:
> E := EllipticCurve([0, -1, 1, 0, 0]);
> Conductor(E);
11
> BadPrimes(E);
[ 11 ]
Hyperelliptic curves and hypergeometric motives
CrvHyp generalises the Weierstrass picture to higher genus: a hyperelliptic curve is given by $y^2 + h(x)y = f(x)$, with specialised support for genus 2 curves over $\Q$ (Jacobian heights, the Kummer surface SrfKum) and for point-counting on Jacobians over finite fields. Separately, HypGeomMot studies hypergeometric motives — arithmetic objects built from the monodromy of generalised hypergeometric differential equations, parametrised by tuples $\vec\alpha, \vec\beta$ of rationals modulo 1, and used to produce explicit motivic L-functions.
L-functions as the common output
Whatever the source object — an elliptic curve, a number field, a Dirichlet character, an Artin representation, a modular form, a hypergeometric motive — its arithmetic content is often packaged as an LSeries. This chapter provides a uniform interface: construct an L-series from the object, then evaluate it, differentiate it, or combine several via sum, product, or tensor product, all against a common functional-equation framework.
> L := LSeries(EllipticCurve([0, -1, 1, 0, 0]));
> Evaluate(L, 2);
0.546048036215013518334126660433
Modular curves, small models, and Fuchsian groups
Modular curves (CrvMod) are plane curves defined by modular polynomials relating the $j$-invariant to a function on $X_0(N)$, mainly used to compute isogenies. The Small Modular Curves database supplies low-degree, sparsely-coefficiented models of $X_0(N)$ together with cusps, automorphisms, and projections between levels, for use when $X_0(N)$ has genus zero or one. Underlying all of this is the group-theoretic side: GrpPSL2 implements congruence subgroups of $\PSL_2(\R)$ acting on the upper half-plane, with fundamental domains via Farey symbols, while GrpPSL2Shim extends the same picture to arithmetic Fuchsian groups coming from quaternion algebras, whose quotients are Shimura curves — a genuine generalisation of the classical modular curve story.
Modular forms, symbols, and Hecke modules
ModFrm is the modular forms package proper: spaces of forms of given weight, level, and character, with Hecke operators and newform decompositions. ModSym implements modular symbols, a homological presentation used under the hood to compute eigenform bases and to connect modularity questions to elliptic curve computations. ModBrdt (Brandt modules) gives an alternative, quaternion-ideal-based construction of the same Hecke-module structure, generalising to Shimura curves $X_0^D(N)$; ModSS specialises this to divisors on supersingular points of $X_0(N)$ in characteristic $p$, a construction historically tied to graphs of supersingular elliptic curves.
> M := ModularForms(11, 2);
> Dimension(M);
2
Modular abelian varieties and their generalisations
ModAbVar represents modular abelian varieties — quotients of $J_1(N)$ — computed entirely via modular symbols rather than explicit equations, supporting decomposition, endomorphism rings, and L-function special values. Three further packages extend the notion of "modular form" to richer settings: ModFrmHil (Hilbert modular forms over totally real fields), ModFrmBianchi (Bianchi modular forms over imaginary quadratic fields, functions on hyperbolic 3-space), and ModFrmAlg (algebraic modular forms for orthogonal and unitary groups). All three follow the same template — compute Hecke operators, decompose into eigenspaces, extract eigenvalues — applied to progressively more general ambient groups.
Local representations
Finally, RepLoc closes the loop between modular forms and Galois representations: starting from a cuspidal newform, it constructs the local automorphic representation at a prime $p$ and, via the local Langlands correspondence, the associated Galois representation of the absolute Galois group of $\Q_p$.
Orientation, not exhaustive coverage
None of this replaces the dedicated Elliptic Curves lesson, and none of it requires memorising every intrinsic. The goal is recognition: when a problem involves genus, isogenies, L-functions, or Hecke operators, you should now have a sense of which chapter's type system it belongs to.