Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -2245670064, 40961041011648]); // or
magma: E := EllipticCurve("14490bb6");
sage: E = EllipticCurve([1, -1, 0, -2245670064, 40961041011648]) # or
sage: E = EllipticCurve("14490bb6")
gp: E = ellinit([1, -1, 0, -2245670064, 40961041011648]) \\ or
gp: E = ellinit("14490bb6")

$$y^2 + x y = x^{3} - x^{2} - 2245670064 x + 40961041011648$$

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()

 $$P$$ = $$\left(-\frac{130557}{4}, \frac{71454477}{8}\right)$$ $$\hat{h}(P)$$ ≈ 7.27085192609

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(27312, -13656\right)$$, $$\left(42312, 4636344\right)$$

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(14712, 3325344\right)$$, $$\left(26787, 149619\right)$$, $$\left(27312, -13656\right)$$, $$\left(27408, -13704\right)$$, $$\left(27937, 151344\right)$$, $$\left(42312, 4636344\right)$$

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$14490$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$6692612332320140625000000$$ = $$2^{6} \cdot 3^{10} \cdot 5^{12} \cdot 7^{2} \cdot 23^{6}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{1718043013877225552292911401729}{9180538178765625000000}$$ = $$2^{-6} \cdot 3^{-4} \cdot 5^{-12} \cdot 7^{-2} \cdot 23^{-6} \cdot 31^{3} \cdot 193^{3} \cdot 733^{3} \cdot 2731^{3}$$ 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: $$7.27085192609$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.066484157776$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1152$$  = $$2\cdot2^{2}\cdot( 2^{2} \cdot 3 )\cdot2\cdot( 2 \cdot 3 )$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$12$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 14490.2.a.z

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
7962624 . This curve is not $$\Gamma_0(N)$$-optimal.

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/factorial(ar)

$$L'(E,1)$$ ≈ $$3.86717173296$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$4$$ $$I_4^{*}$$ Additive -1 2 10 4
$$5$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$23$$ $$6$$ $$I_{6}$$ 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 ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary 5 - 2 4 1,1 1 1 1 2 1 1 1 1 1 1 1 - 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 14490.z 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 $$\Q(\sqrt{-6}, \sqrt{-14})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-21}, \sqrt{-69})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{6}, \sqrt{46})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.425329947.3 $$\Z/6\Z \times \Z/6\Z$$ 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.