Properties

Label 90.c2
Conductor 90
Discriminant 4271484375000
j-invariant \( \frac{10316097499609}{5859375000} \)
CM no
Rank 0
Torsion Structure \(\Z/{6}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -4082, 14681]); // or
magma: E := EllipticCurve("90c7");
sage: E = EllipticCurve([1, -1, 1, -4082, 14681]) # or
sage: E = EllipticCurve("90c7")
gp: E = ellinit([1, -1, 1, -4082, 14681]) \\ or
gp: E = ellinit("90c7")

\( y^2 + x y + y = x^{3} - x^{2} - 4082 x + 14681 \)

Mordell-Weil group structure

\(\Z/{6}\Z\)

Torsion generators

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

\( \left(-9, 229\right) \)

Integral points

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

\( \left(-9, 229\right) \), \( \left(91, 579\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 90 \)  =  \(2 \cdot 3^{2} \cdot 5\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(4271484375000 \)  =  \(2^{3} \cdot 3^{7} \cdot 5^{12} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{10316097499609}{5859375000} \)  =  \(2^{-3} \cdot 3^{-1} \cdot 5^{-12} \cdot 11^{3} \cdot 1979^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(0\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.668797997294\)
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: \( 72 \)  = \( 3\cdot2\cdot( 2^{2} \cdot 3 ) \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
Torsion order: \(6\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 90.2.a.c

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)

\( q + q^{2} + q^{4} + q^{5} - 4q^{7} + q^{8} + q^{10} + 2q^{13} - 4q^{14} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
192 . This curve is not \( \Gamma_0(N) \)-optimal.

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])

\( L(E,1) \) ≈ \( 1.33759599459 \)

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\((j)_{-}\)
\(2\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(3\) \(2\) \( I_1^{*} \) Additive -1 2 7 1
\(5\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12

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 X13f.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 12.

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.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]

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5
Reduction type split add split
$\lambda$-invariant(s) 1 - 1
$\mu$-invariant(s) 1 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 90.c 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{6}) \) \(\Z/2\Z \times \Z/6\Z\) 2.2.24.1-150.1-b5
\(\Q(\sqrt{-6}) \) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-1}) \) \(\Z/12\Z\) 2.0.4.1-4050.2-c3
4 4.2.55296.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
\(\Q(i, \sqrt{6})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
6 6.0.177147.2 \(\Z/3\Z \times \Z/6\Z\) 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.