Properties

 Label 6930.o1 Conductor 6930 Discriminant 46430323242187500 j-invariant $$\frac{1969902499564819009}{63690429687500}$$ CM no Rank 1 Torsion Structure $$\Z/{6}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -235044, 42676308]); // or

magma: E := EllipticCurve("6930p4");

sage: E = EllipticCurve([1, -1, 0, -235044, 42676308]) # or

sage: E = EllipticCurve("6930p4")

gp: E = ellinit([1, -1, 0, -235044, 42676308]) \\ or

gp: E = ellinit("6930p4")

$$y^2 + x y = x^{3} - x^{2} - 235044 x + 42676308$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(22, 6114\right)$$ $$\hat{h}(P)$$ ≈ 1.15934793426

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-153, 8739\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-531, 4455\right)$$, $$\left(-531, -3924\right)$$, $$\left(-153, 8739\right)$$, $$\left(-153, -8586\right)$$, $$\left(22, 6114\right)$$, $$\left(22, -6136\right)$$, $$\left(232, 654\right)$$, $$\left(232, -886\right)$$, $$\left(342, 1314\right)$$, $$\left(342, -1656\right)$$, $$\left(397, 3239\right)$$, $$\left(397, -3636\right)$$, $$\left(522, 7614\right)$$, $$\left(522, -8136\right)$$, $$\left(7272, 615114\right)$$, $$\left(7272, -622386\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$6930$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$46430323242187500$$ = $$2^{2} \cdot 3^{6} \cdot 5^{12} \cdot 7^{2} \cdot 11^{3}$$ 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: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.15934793426$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.356662443493$$ 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: $$288$$  = $$2\cdot2\cdot( 2^{2} \cdot 3 )\cdot2\cdot3$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$6$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form6930.2.a.o

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 82944 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$3.30796693673$$

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
$$5$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 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.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 ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 - 2 2 2 1 1,1 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.o 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{11})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4 4.0.19404.2 $$\Z/12\Z$$ Not in database
6 6.0.84015792.1 $$\Z/3\Z \times \Z/6\Z$$ 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.