# Properties

 Label 10710.n4 Conductor 10710 Discriminant 41921259685871942250000 j-invariant $$\frac{116454264690812369959009}{57505157319440250000}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -9156294, 4087006308]); // or

magma: E := EllipticCurve("10710l6");

sage: E = EllipticCurve([1, -1, 0, -9156294, 4087006308]) # or

sage: E = EllipticCurve("10710l6")

gp: E = ellinit([1, -1, 0, -9156294, 4087006308]) \\ or

gp: E = ellinit("10710l6")

$$y^2 + x y = x^{3} - x^{2} - 9156294 x + 4087006308$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z \times \Z/{6}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-3143, 44114\right)$$ $$\hat{h}(P)$$ ≈ 3.12397433743

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(2772, -1386\right)$$, $$\left(-1953, 121464\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-3228, 1614\right)$$, $$\left(-3143, 44114\right)$$, $$\left(-3143, -40971\right)$$, $$\left(-2412, 111366\right)$$, $$\left(-2412, -108954\right)$$, $$\left(-1953, 121464\right)$$, $$\left(-1953, -119511\right)$$, $$\left(-168, 75054\right)$$, $$\left(-168, -74886\right)$$, $$\left(2772, -1386\right)$$, $$\left(3997, 175014\right)$$, $$\left(3997, -179011\right)$$, $$\left(4872, 271614\right)$$, $$\left(4872, -276486\right)$$, $$\left(9387, 858564\right)$$, $$\left(9387, -867951\right)$$, $$\left(14112, 1631574\right)$$, $$\left(14112, -1645686\right)$$, $$\left(256872, 130051614\right)$$, $$\left(256872, -130308486\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$10710$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$41921259685871942250000$$ = $$2^{4} \cdot 3^{10} \cdot 5^{6} \cdot 7^{6} \cdot 17^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{116454264690812369959009}{57505157319440250000}$$ = $$2^{-4} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-6} \cdot 17^{-6} \cdot 73^{3} \cdot 307^{3} \cdot 2179^{3}$$ 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: $$3.12397433743$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.101478789785$$ 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: $$1728$$  = $$2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$12$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 10710.2.a.n

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)

$$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2q^{13} - q^{14} + q^{16} + q^{17} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 884736 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$3.804205621$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$4$$ $$I_4^{*}$$ Additive -1 2 10 4
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$7$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$17$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6

## 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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

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
$$2$$ Cs
$$3$$ B.1.1

## $p$-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 nonsplit add split split ss ordinary split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 7 - 2 2 1,1 1 2 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

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, 3 and 6.
Its isogeny class 10710.n consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-15}, \sqrt{-51})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{15}, \sqrt{21})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-21}, \sqrt{51})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.2834352.2 $$\Z/6\Z \times \Z/6\Z$$ 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.