# Properties

 Label 381150.ev1 Conductor 381150 Discriminant 1285221091769577026367187500 j-invariant $$\frac{1969902499564819009}{63690429687500}$$ CM no Rank 2 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -711008667, -7090316624759]); // or

magma: E := EllipticCurve("381150ev4");

sage: E = EllipticCurve([1, -1, 0, -711008667, -7090316624759]) # or

sage: E = EllipticCurve("381150ev4")

gp: E = ellinit([1, -1, 0, -711008667, -7090316624759]) \\ or

gp: E = ellinit("381150ev4")

$$y^2 + x y = x^{3} - x^{2} - 711008667 x - 7090316624759$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-14682, -421241\right)$$ $$\left(-\frac{2372124}{169}, \frac{789246931}{2197}\right)$$ $$\hat{h}(P)$$ ≈ 4.99420141033 6.90258007823

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-\frac{52909}{4}, \frac{52909}{8}\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-14682, 435923\right)$$, $$\left(-14682, -421241\right)$$, $$\left(-13351, 156413\right)$$, $$\left(-13351, -143062\right)$$, $$\left(71924, 17679413\right)$$, $$\left(71924, -17751337\right)$$, $$\left(216824, 100052363\right)$$, $$\left(216824, -100269187\right)$$, $$\left(373035, 227051906\right)$$, $$\left(373035, -227424941\right)$$, $$\left(1490679, 1818981223\right)$$, $$\left(1490679, -1820471902\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$381150$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$1285221091769577026367187500$$ = $$2^{2} \cdot 3^{6} \cdot 5^{18} \cdot 7^{2} \cdot 11^{9}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{1969902499564819009}{63690429687500}$$ = $$2^{-2} \cdot 5^{-12} \cdot 7^{-2} \cdot 11^{-3} \cdot 23^{3} \cdot 54503^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$12.9486694686$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0292948389314$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$128$$  = $$2\cdot2\cdot2^{2}\cdot2\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 381150.2.a.ev

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{2} + q^{4} + q^{7} - q^{8} - 4q^{13} - q^{14} + q^{16} + 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 238878720 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$12.1385339667$$

## Local data

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

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$5$$ $$4$$ $$I_12^{*}$$ Additive 1 2 18 12
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$11$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3

## 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.

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
$$2$$ B
$$3$$ 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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 381150.ev consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $$\Q(\sqrt{11})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-55})$$ $$\Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-5}, \sqrt{11})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4.0.485100.1 $$\Z/4\Z$$ Not in database
6 6.2.13978127394000.2 $$\Z/6\Z$$ 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.