Properties

 Label 6630.v4 Conductor 6630 Discriminant 8834417507562311995200 j-invariant $$\frac{59589391972023341137821784609}{8834417507562311995200}$$ 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, 0, 0, -81373266, -282504599100]); // or

magma: E := EllipticCurve("6630w4");

sage: E = EllipticCurve([1, 0, 0, -81373266, -282504599100]) # or

sage: E = EllipticCurve("6630w4")

gp: E = ellinit([1, 0, 0, -81373266, -282504599100]) \\ or

gp: E = ellinit("6630w4")

$$y^2 + x y = x^{3} - 81373266 x - 282504599100$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-5166, 5490\right)$$ $$\hat{h}(P)$$ ≈ 5.06801887944

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-5184, 7794\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-5184, 7794\right)$$, $$\left(-5184, -2610\right)$$, $$\left(-5166, 5490\right)$$, $$\left(-5166, -324\right)$$, $$\left(15624, 1495566\right)$$, $$\left(15624, -1511190\right)$$, $$\left(26606, 4034720\right)$$, $$\left(26606, -4061326\right)$$, $$\left(213300, 98315190\right)$$, $$\left(213300, -98528490\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$6630$$ = $$2 \cdot 3 \cdot 5 \cdot 13 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$8834417507562311995200$$ = $$2^{6} \cdot 3^{6} \cdot 5^{2} \cdot 13 \cdot 17^{12}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{59589391972023341137821784609}{8834417507562311995200}$$ = $$2^{-6} \cdot 3^{-6} \cdot 5^{-2} \cdot 13^{-1} \cdot 17^{-12} \cdot 359^{3} \cdot 433^{3} \cdot 25127^{3}$$ 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: $$5.06801887944$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0502670016953$$ 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: $$864$$  = $$( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot1\cdot( 2^{2} \cdot 3 )$$ 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 form6630.2.a.v

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1105920 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$6.11409872652$$

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$$(j)_{-}$$
$$2$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$17$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12

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 X13f.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 12.

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 split split nonsplit ordinary ss split split ordinary ss ordinary ordinary ordinary ordinary ordinary ss 3 4 1 1 1,1 2 2 1 1,1 1 1 1 1 3 1,1 2 0 0 0 0,0 0 0 0 0,0 0 0 0 0 0 0,0

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 6630.v 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-1})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{13})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{-13})$$ $$\Z/12\Z$$ Not in database
4 4.2.31636800.1 $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(i, \sqrt{13})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.481966875.1 $$\Z/3\Z \times \Z/6\Z$$ 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.