# Properties

 Label 40310bd1 Conductor 40310 Discriminant -239773280695100000000000000 j-invariant $$-\frac{107350761560343751123953328881}{239773280695100000000000000}$$ CM no Rank 0 Torsion Structure $$\Z/{7}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -99013362, -835939931151]) # or

sage: E = EllipticCurve("40310bd1")

gp: E = ellinit([1, -1, 1, -99013362, -835939931151]) \\ or

gp: E = ellinit("40310bd1")

magma: E := EllipticCurve([1, -1, 1, -99013362, -835939931151]); // or

magma: E := EllipticCurve("40310bd1");

$$y^2 + x y + y = x^{3} - x^{2} - 99013362 x - 835939931151$$

## Mordell-Weil group structure

$$\Z/{7}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(13857, 665871\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(13857, 665871\right)$$, $$\left(13857, -679729\right)$$, $$\left(25457, 3612271\right)$$, $$\left(25457, -3637729\right)$$, $$\left(97957, 30437271\right)$$, $$\left(97957, -30535229\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$40310$$ = $$2 \cdot 5 \cdot 29 \cdot 139$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-239773280695100000000000000$$ = $$-1 \cdot 2^{14} \cdot 5^{14} \cdot 29^{7} \cdot 139$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{107350761560343751123953328881}{239773280695100000000000000}$$ = $$-1 \cdot 2^{-14} \cdot 3^{3} \cdot 5^{-14} \cdot 13^{3} \cdot 19^{3} \cdot 29^{-7} \cdot 139^{-1} \cdot 6413821^{3}$$ 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[1]  magma: RealPeriod(E); Real period: $$0.0223902292855$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1372$$  = $$( 2 \cdot 7 )\cdot( 2 \cdot 7 )\cdot7\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$7$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 40310.2.a.p

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 29503488 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L(E,1)$$ ≈ $$0.626926419994$$

## Local data

This elliptic curve is semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$5$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$29$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$139$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

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

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
$$7$$ 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]

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 7 29 139 split ss split ordinary split nonsplit 7 0,0 1 4 1 0 0 0,0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 40310bd consists of 2 curves linked by isogenies of degree 7.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.4031.1 $$\Z/14\Z$$ Not in database
6 6.0.65499561791.1 $$\Z/2\Z \times \Z/14\Z$$ 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.