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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -11979, -510867]) # or

sage: E = EllipticCurve("90354b1")

gp: E = ellinit([1, 1, 0, -11979, -510867]) \\ or

gp: E = ellinit("90354b1")

magma: E := EllipticCurve([1, 1, 0, -11979, -510867]); // or

magma: E := EllipticCurve("90354b1");

$$y^2 + x y = x^{3} + x^{2} - 11979 x - 510867$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(126, -63\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(126, -63\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$90354$$ = $$2 \cdot 3 \cdot 11 \cdot 37^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-526417583472$$ = $$-1 \cdot 2^{4} \cdot 3^{10} \cdot 11 \cdot 37^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3753503985421}{10392624}$$ = $$-1 \cdot 2^{-4} \cdot 3^{-10} \cdot 11^{-1} \cdot 15541^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.228132770881$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$8$$  = $$2\cdot2\cdot1\cdot2$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$4$$ (exact)

## Modular invariants

#### Modular form 90354.2.a.e

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} - 2q^{10} - q^{11} - q^{12} + 6q^{13} + 2q^{14} - 2q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 230400 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

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

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

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

$$L(E,1)$$ ≈ $$1.82506216705$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$I_{10}$$ Non-split multiplicative 1 1 10 10
$$11$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$37$$ $$2$$ $$III$$ Additive -1 2 3 0

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

sage: rho = E.galois_representation();

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

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

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B
$$5$$ B

## $p$-adic data

### $p$-adic regulators

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

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 37 nonsplit nonsplit ordinary nonsplit add 2 0 0 0 - 0 0 0 0 -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

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, 5 and 10.
Its isogeny class 90354b consists of 4 curves linked by isogenies of degrees dividing 10.

## 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 $$\Q(\sqrt{-407})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.0.50653.1 $$\Z/10\Z$$ Not in database
4.2.80234352.3 $$\Z/4\Z$$ 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.