Properties

 Label 40950.dh5 Conductor 40950 Discriminant 249394337075871146250000 j-invariant $$\frac{443915739051786565201}{21894701746029840}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -35758130, 78725614497]); // or

magma: E := EllipticCurve("40950dm3");

sage: E = EllipticCurve([1, -1, 1, -35758130, 78725614497]) # or

sage: E = EllipticCurve("40950dm3")

gp: E = ellinit([1, -1, 1, -35758130, 78725614497]) \\ or

gp: E = ellinit("40950dm3")

$$y^2 + x y + y = x^{3} - x^{2} - 35758130 x + 78725614497$$

Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(2739, 35055\right)$$ $$\hat{h}(P)$$ ≈ 5.62801432055

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-6871, 3435\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-6871, 3435\right)$$, $$\left(2739, 35055\right)$$

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

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$40950$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$249394337075871146250000$$ = $$2^{4} \cdot 3^{8} \cdot 5^{7} \cdot 7^{12} \cdot 13^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{443915739051786565201}{21894701746029840}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-3} \cdot 37^{3} \cdot 18743^{3}$$ 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: $$5.62801432055$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0973703808041$$ 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: $$64$$  = $$2^{2}\cdot2\cdot2^{2}\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 40950.2.a.dh

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 5308416 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$8.76803036101$$

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$3$$ $$2$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$7$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$13$$ $$1$$ $$I_{3}$$ Non-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 X13.

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

$p$-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 add add nonsplit ss nonsplit ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss 5 - - 1 1,1 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, 4, 6 and 12.
Its isogeny class 40950.dh 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-39})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{5})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{-15})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{65})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$\Q(\sqrt{5}, \sqrt{13})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{5}, \sqrt{-39})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{5})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{-15}, \sqrt{-39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 6.0.8857350000.7 $$\Z/12\Z$$ 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.