Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, 0, 12]); // or

magma: E := EllipticCurve("441b1");

sage: E = EllipticCurve([0, 0, 1, 0, 12]) # or

sage: E = EllipticCurve("441b1")

gp: E = ellinit([0, 0, 1, 0, 12]) \\ or

gp: E = ellinit("441b1")

$$y^2 + y = x^{3} + 12$$

## Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(2, 4\right)$$ $$\hat{h}(P)$$ ≈ 0.894275780214

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(0, 3\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(0, 3\right)$$, $$\left(0, -4\right)$$, $$\left(2, 4\right)$$, $$\left(2, -5\right)$$, $$\left(14, 52\right)$$, $$\left(14, -53\right)$$

## 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: $$-64827$$ = $$-1 \cdot 3^{3} \cdot 7^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$0$$ = $$0$$ Endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ ( 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: $$0.894275780214$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$2.77057339991$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$6$$  = $$2\cdot3$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$3$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form441.2.a.d

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - 2q^{4} - 7q^{13} + 4q^{16} - 7q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 24 $$\Gamma_0(N)$$-optimal: yes 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)

$$L'(E,1)$$ ≈ $$1.6517711259$$

## 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$$(j)_{-}$$
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$7$$ $$3$$ $$IV$$ Additive 1 2 4 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
$$3$$ B.1.1
$$7$$ Ns.6.1.2

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 ss add ss add ss ordinary ss ordinary ss ss ordinary ordinary ss ordinary ss ? - 1,1 - 1,1 1 1,1 1 1,1 1,1 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 $$d=$$ 3.
Its isogeny class 441.d consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.588.1 $$\Z/6\Z$$ Not in database
6 6.0.5250987.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.1037232.1 $$\Z/2\Z \times \Z/6\Z$$ 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.