Properties

 Label 42042dk2 Conductor 42042 Discriminant 444791982921302016 j-invariant $$\frac{28826282175168869972161}{9077387406557184}$$ CM no Rank 1 Torsion Structure $$\Z/{7}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -2337511, 1374987977]) # or

sage: E = EllipticCurve("42042dk2")

gp: E = ellinit([1, 0, 0, -2337511, 1374987977]) \\ or

gp: E = ellinit("42042dk2")

magma: E := EllipticCurve([1, 0, 0, -2337511, 1374987977]); // or

magma: E := EllipticCurve("42042dk2");

$$y^2 + x y = x^{3} - 2337511 x + 1374987977$$

Mordell-Weil group structure

$$\Z\times \Z/{7}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1738, 14597\right)$$ $$\hat{h}(P)$$ ≈ 2.5004240694637696

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(902, 341\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1738, 14597\right)$$, $$\left(-1738, -12859\right)$$, $$\left(-286, 45089\right)$$, $$\left(-286, -44803\right)$$, $$\left(866, 161\right)$$, $$\left(866, -1027\right)$$, $$\left(902, 341\right)$$, $$\left(902, -1243\right)$$, $$\left(1034, 7469\right)$$, $$\left(1034, -8503\right)$$, $$\left(2486, 103301\right)$$, $$\left(2486, -105787\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$42042$$ = $$2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 13$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$444791982921302016$$ = $$2^{14} \cdot 3^{7} \cdot 7^{2} \cdot 11^{7} \cdot 13$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{28826282175168869972161}{9077387406557184}$$ = $$2^{-14} \cdot 3^{-7} \cdot 7 \cdot 11^{-7} \cdot 13^{-1} \cdot 19^{3} \cdot 843613^{3}$$ 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: $$2.50042406946$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.290926865768$$ 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: $$686$$  = $$( 2 \cdot 7 )\cdot7\cdot1\cdot7\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$7$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 42042.2.a.db

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);

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - q^{13} - q^{15} + q^{16} - 3q^{17} + q^{18} - q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1119552 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$10.1841675267$$

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$$(j)_{-}$$
$$2$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$11$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$13$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$7$$ 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 ordinary add split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary 3 2 1 - 2 3 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

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 $$d=$$ 7.
Its isogeny class 42042dk consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.21021.1 $$\Z/14\Z$$ Not in database
6 6.6.189567567189.1 $$\Z/2\Z \times \Z/14\Z$$ 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.