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Properties

Label 40425.p2
Conductor 40425
Discriminant 162151572515625
j-invariant \( \frac{169112377}{88209} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -14113, -208594]); // or
magma: E := EllipticCurve("40425y2");
sage: E = EllipticCurve([1, 1, 1, -14113, -208594]) # or
sage: E = EllipticCurve("40425y2")
gp: E = ellinit([1, 1, 1, -14113, -208594]) \\ or
gp: E = ellinit("40425y2")

\( y^2 + x y + y = x^{3} + x^{2} - 14113 x - 208594 \)

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()

\(P\) =  \( \left(-64, 693\right) \)
\(\hat{h}(P)\) ≈  1.30634595814

Torsion generators

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

\( \left(-15, 7\right) \), \( \left(125, -63\right) \)

Integral points

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

\( \left(-64, 693\right) \), \( \left(-50, 637\right) \), \( \left(-15, 7\right) \), \( \left(125, -63\right) \), \( \left(129, 307\right) \), \( \left(260, 3582\right) \), \( \left(370, 6552\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 40425 \)  =  \(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(162151572515625 \)  =  \(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{2} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{169112377}{88209} \)  =  \(3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}\)
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: \(1.30634595814\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.463927711946\)
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: \( 64 \)  = \( 2\cdot2^{2}\cdot2^{2}\cdot2 \)
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 40425.2.a.p

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)

\( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + q^{11} + q^{12} - 2q^{13} - q^{16} - 2q^{17} - q^{18} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 110592
\( \Gamma_0(N) \)-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])

\( L'(E,1) \) ≈ \( 2.42420036547 \)

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\((j)_{-}\)
\(3\) \(2\) \( I_{6} \) Non-split multiplicative 1 1 6 6
\(5\) \(4\) \( I_0^{*} \) Additive 1 2 6 0
\(7\) \(4\) \( I_0^{*} \) Additive -1 2 6 0
\(11\) \(2\) \( I_{2} \) 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

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ordinary nonsplit add add split ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss ordinary
$\lambda$-invariant(s) 8 1 - - 2 3 1 1,1 1 1 1 1 1 1,1 1
$\mu$-invariant(s) 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.
Its isogeny class 40425.p consists of 4 curves linked by isogenies of degrees dividing 4.

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
4 \(\Q(\sqrt{3}, \sqrt{35})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{33}, \sqrt{-105})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-11}, \sqrt{-35})\) \(\Z/2\Z \times \Z/4\Z\) 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.