Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -16400, -804212]); // or

magma: E := EllipticCurve("441d4");

sage: E = EllipticCurve([1, -1, 1, -16400, -804212]) # or

sage: E = EllipticCurve("441d4")

gp: E = ellinit([1, -1, 1, -16400, -804212]) \\ or

gp: E = ellinit("441d4")

$$y^2 + x y + y = x^{3} - x^{2} - 16400 x - 804212$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-74, 41\right)$$ $$\hat{h}(P)$$ ≈ 1.53426186827

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-\frac{293}{4}, \frac{289}{8}\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-74, 41\right)$$, $$\left(-74, 32\right)$$, $$\left(184, 1451\right)$$, $$\left(184, -1636\right)$$, $$\left(389, 6986\right)$$, $$\left(389, -7376\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$441$$ = $$3^{2} \cdot 7^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$29417779503$$ = $$3^{6} \cdot 7^{9}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$16581375$$ = $$3^{3} \cdot 5^{3} \cdot 17^{3}$$ Endomorphism ring: $$\Z[\sqrt{-7}]$$ ( Complex Multiplication) Sato-Tate Group: $N(\mathrm{U}(1))$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.53426186827$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.421883201568$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$8$$  = $$2^{2}\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form441.2.a.c

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - q^{2} - q^{4} + 3q^{8} - 4q^{11} - q^{16} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 448 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

$$L'(E,1)$$ ≈ $$1.29455861806$$

## Local data

This elliptic curve is not semistable.

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$7$$ $$2$$ $$III^{*}$$ Additive -1 2 9 0

## Galois representations

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois representation
$$7$$ B.6.2

For all other primes $$p$$, the image is a Borel subgroup if $$p=2$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=-1$$.

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ordinary add ss add ordinary ss ss ss ordinary ordinary ss ordinary ss ordinary ss ? - 1,1 - 1 1,1 1,1 1,1 1 1 3,1 1 1,1 1 1,1 ? - 0,0 - 0 0,0 0,0 0,0 0 0 0,0 0 0,0 0 0,0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 7 and 14.
Its isogeny class 441.c consists of 4 curves linked by isogenies of degrees dividing 14.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{7})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.0.12348.1 $$\Z/4\Z$$ Not in database
6 6.0.64827.1 $$\Z/14\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.