Properties

Label 90354c1
Conductor 90354
Discriminant -168830437009509039006
j-invariant \( \frac{1298596571}{1299078} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 1151301, 406350531]) # or
 
sage: E = EllipticCurve("90354c1")
 
gp: E = ellinit([1, 1, 0, 1151301, 406350531]) \\ or
 
gp: E = ellinit("90354c1")
 
magma: E := EllipticCurve([1, 1, 0, 1151301, 406350531]); // or
 
magma: E := EllipticCurve("90354c1");
 

\( y^2 + x y = x^{3} + x^{2} + 1151301 x + 406350531 \)

Mordell-Weil group structure

Trivial

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 
None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 90354 \)  =  \(2 \cdot 3 \cdot 11 \cdot 37^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-168830437009509039006 \)  =  \(-1 \cdot 2 \cdot 3^{10} \cdot 11 \cdot 37^{9} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1298596571}{1299078} \)  =  \(2^{-1} \cdot 3^{-10} \cdot 11^{-1} \cdot 1091^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.119337977471\)
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: \( 4 \)  = \( 1\cdot2\cdot1\cdot2 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 90354.2.a.b

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);
 

\( q - q^{2} - q^{3} + q^{4} - 3q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 3q^{10} - q^{11} - q^{12} - 4q^{13} + 2q^{14} + 3q^{15} + q^{16} + 3q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3729600
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L(E,1) \) ≈ \( 0.477351909886 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(3\) \(2\) \( I_{10} \) Non-split multiplicative 1 1 10 10
\(11\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(37\) \(2\) \( III^{*} \) Additive -1 2 9 0

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
\(5\) B

$p$-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 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit nonsplit ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss add ordinary ordinary ordinary
$\lambda$-invariant(s) 8 0 0 0 0 0 0 0 0 0 0,0 - 0 0 0
$\mu$-invariant(s) 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 \(d=\) 5.
Its isogeny class 90354c consists of 2 curves linked by isogenies of degree 5.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.3256.1 \(\Z/2\Z\) Not in database
4 4.0.50653.1 \(\Z/5\Z\) Not in database
6 6.0.34518601216.1 \(\Z/2\Z \times \Z/2\Z\) 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.