Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 7412, 212781]); // or

magma: E := EllipticCurve("550i2");

sage: E = EllipticCurve([1, 1, 1, 7412, 212781]) # or

sage: E = EllipticCurve("550i2")

gp: E = ellinit([1, 1, 1, 7412, 212781]) \\ or

gp: E = ellinit("550i2")

$$y^2 + x y + y = x^{3} + x^{2} + 7412 x + 212781$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(5, 497\right)$$ $$\hat{h}(P)$$ ≈ 0.0642004440205

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-19, 265\right)$$, $$\left(5, 497\right)$$, $$\left(45, 777\right)$$, $$\left(245, 3977\right)$$, $$\left(309, 5513\right)$$, $$\left(1005, 31497\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$550$$ = $$2 \cdot 5^{2} \cdot 11$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-45056000000000$$ = $$-1 \cdot 2^{21} \cdot 5^{9} \cdot 11$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{2882081488391}{2883584000}$$ = $$2^{-21} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-1} \cdot 19^{3} \cdot 107^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.0642004440205$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.421301130369$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$84$$  = $$( 3 \cdot 7 )\cdot2^{2}\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form550.2.a.i

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + q^{2} - q^{3} + q^{4} - q^{6} - 5q^{7} + q^{8} - 2q^{9} + q^{11} - q^{12} - 2q^{13} - 5q^{14} + q^{16} - 3q^{17} - 2q^{18} - 7q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 2016 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$2.27200844943$$

## Local data

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$$(j)_{-}$$
$$2$$ $$21$$ $$I_{21}$$ Split multiplicative -1 1 21 21
$$5$$ $$4$$ $$I_3^{*}$$ Additive 1 2 9 3
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-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 split ordinary add ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 3 1 - 1 4 1 1 1 1 1 3 1 1 1 1 0 1 - 0 0 0 0 0 0 0 0 0 0 0 0

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 550.i 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-15})$$ $$\Z/3\Z$$ Not in database
3 3.1.440.1 $$\Z/2\Z$$ Not in database
6 6.0.85184000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.0.26136000.1 $$\Z/6\Z$$ Not in database
6.2.1334161125.3 $$\Z/3\Z$$ 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.