Properties

 Label 6930.q3 Conductor 6930 Discriminant 35596085895562500 j-invariant $$\frac{4074571110566294433649}{48828650062500}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -2994759, 1995489513]) # or

sage: E = EllipticCurve("6930o6")

gp: E = ellinit([1, -1, 0, -2994759, 1995489513]) \\ or

gp: E = ellinit("6930o6")

magma: E := EllipticCurve([1, -1, 0, -2994759, 1995489513]); // or

magma: E := EllipticCurve("6930o6");

$$y^2 + x y = x^{3} - x^{2} - 2994759 x + 1995489513$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1833, 37299\right)$$ $$\hat{h}(P)$$ ≈ 2.5503935441240095

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1002, -501\right)$$, $$\left(477, 25749\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1998, 999\right)$$, $$\left(-1833, 37299\right)$$, $$\left(-1833, -35466\right)$$, $$\left(202, 37299\right)$$, $$\left(202, -37501\right)$$, $$\left(477, 25749\right)$$, $$\left(477, -26226\right)$$, $$\left(664, 16971\right)$$, $$\left(664, -17635\right)$$, $$\left(972, 999\right)$$, $$\left(972, -1971\right)$$, $$\left(1002, -501\right)$$, $$\left(1027, 999\right)$$, $$\left(1027, -2026\right)$$, $$\left(1044, 1935\right)$$, $$\left(1044, -2979\right)$$, $$\left(1377, 21249\right)$$, $$\left(1377, -22626\right)$$, $$\left(1632, 37299\right)$$, $$\left(1632, -38931\right)$$, $$\left(52452, 11979999\right)$$, $$\left(52452, -12032451\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$6930$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$35596085895562500$$ = $$2^{2} \cdot 3^{8} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4074571110566294433649}{48828650062500}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-6} \cdot 37^{3} \cdot 43^{3} \cdot 10039^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.55039354412$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.333155767767$$ 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: $$576$$  = $$2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 )$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$12$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form6930.2.a.q

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 165888 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$3.39871327721$$

Local data

This elliptic curve is not 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$11$$ $$6$$ $$I_{6}$$ 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.

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
$$2$$ 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 split ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss 4 - 2 2 2 1 1 3 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 6930.q 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 $$\Q(\sqrt{11}, \sqrt{-21})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{21}, \sqrt{30})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-11}, \sqrt{-30})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.6805279152.4 $$\Z/6\Z \times \Z/6\Z$$ 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.