Properties

 Label 40560ca6 Conductor 40560 Discriminant 3242874250137600 j-invariant $$\frac{272223782641}{164025}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z \times \Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -365096, 84744180]); // or

magma: E := EllipticCurve("40560ca6");

sage: E = EllipticCurve([0, 1, 0, -365096, 84744180]) # or

sage: E = EllipticCurve("40560ca6")

gp: E = ellinit([0, 1, 0, -365096, 84744180]) \\ or

gp: E = ellinit("40560ca6")

$$y^2 = x^{3} + x^{2} - 365096 x + 84744180$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(4, 9126\right)$$ $$\hat{h}(P)$$ ≈ 0.895869619281

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(342, 0\right)$$, $$\left(355, 0\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-698, 0\right)$$, $$\left(4, 9126\right)$$, $$\left(316, 1014\right)$$, $$\left(342, 0\right)$$, $$\left(355, 0\right)$$, $$\left(382, 1080\right)$$, $$\left(391, 1386\right)$$, $$\left(862, 20280\right)$$, $$\left(74302, 20253000\right)$$

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

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$40560$$ = $$2^{4} \cdot 3 \cdot 5 \cdot 13^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$3242874250137600$$ = $$2^{12} \cdot 3^{8} \cdot 5^{2} \cdot 13^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{272223782641}{164025}$$ = $$3^{-8} \cdot 5^{-2} \cdot 6481^{3}$$ 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: $$0.895869619281$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.44271793692$$ 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: $$256$$  = $$2^{2}\cdot2^{3}\cdot2\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$4$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 40560.2.a.bv

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)

$$q + q^{3} - q^{5} + q^{9} - 4q^{11} - q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 294912 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$6.34588079355$$

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_4^{*}$$ Additive -1 4 12 0
$$3$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0

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 X185.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 6 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right)$ and has index 48.

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

$p$-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 add split nonsplit ss ordinary add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary - 2 3 3,1 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, 4 and 8.
Its isogeny class 40560ca consists of 8 curves linked by isogenies of degrees dividing 16.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
4 $$\Q(\sqrt{5}, \sqrt{-13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{10}, \sqrt{13})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database
$$\Q(\sqrt{-5}, \sqrt{-13})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.