Properties

 Label 46410be2 Conductor 46410 Discriminant 261806444735062500 j-invariant $$\frac{5877491705974396839241}{261806444735062500}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{6}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -375973, 85217756]); // or
magma: E := EllipticCurve("46410be2");
sage: E = EllipticCurve([1, 0, 1, -375973, 85217756]) # or
sage: E = EllipticCurve("46410be2")
gp: E = ellinit([1, 0, 1, -375973, 85217756]) \\ or
gp: E = ellinit("46410be2")

$$y^2 + x y + y = x^{3} - 375973 x + 85217756$$

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()

 $$P$$ = $$\left(-\frac{2645}{4}, \frac{56191}{8}\right)$$ $$\hat{h}(P)$$ ≈ 1.67415986853

Torsion generators

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

$$\left(295, -148\right)$$, $$\left(1570, 57227\right)$$

Integral points

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

$$\left(-705, 352\right)$$, $$\left(-380, 13352\right)$$, $$\left(-215, 12602\right)$$, $$\left(-68, 10544\right)$$, $$\left(142, 5819\right)$$, $$\left(205, 3992\right)$$, $$\left(295, -148\right)$$, $$\left(415, 632\right)$$, $$\left(465, 3082\right)$$, $$\left(520, 5252\right)$$, $$\left(1045, 28352\right)$$, $$\left(1570, 57227\right)$$, $$\left(2725, 137552\right)$$, $$\left(24775, 3886052\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$46410$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$261806444735062500$$ = $$2^{2} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 13^{2} \cdot 17^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{5877491705974396839241}{261806444735062500}$$ = $$2^{-2} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{-2} \cdot 17^{-2} \cdot 31^{3} \cdot 79^{3} \cdot 7369^{3}$$ 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: $$1.67415986853$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.307234392232$$ 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: $$1728$$  = $$2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\cdot2$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$12$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 46410.2.a.be

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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 608256 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$6.17231387648$$

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
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$7$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$13$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

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 split split split ss split nonsplit ordinary ss ordinary ordinary ordinary ordinary ordinary ss 7 2 2 2 1,1 2 1 1 1,1 3 1 1 1 1 1,1 0 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 and 6.
Its isogeny class 46410be 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 $$\Q(\sqrt{10}, \sqrt{-51})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-10}, \sqrt{-91})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{51}, \sqrt{91})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.1030511497392.2 $$\Z/6\Z \times \Z/6\Z$$ 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.