Properties

Label 1290.f1
Conductor $1290$
Discriminant $-4.963\times 10^{27}$
j-invariant \( \frac{192203697666261893287480365959}{4963160303408775168000000000} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 120229952, -3351306510322])
 
gp: E = ellinit([1, 0, 1, 120229952, -3351306510322])
 
magma: E := EllipticCurve([1, 0, 1, 120229952, -3351306510322]);
 

\(y^2+xy+y=x^3+120229952x-3351306510322\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1290 \)  =  \(2 \cdot 3 \cdot 5 \cdot 43\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-4963160303408775168000000000 \)  =  \(-1 \cdot 2^{53} \cdot 3^{8} \cdot 5^{9} \cdot 43 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{192203697666261893287480365959}{4963160303408775168000000000} \)  =  \(2^{-53} \cdot 3^{-8} \cdot 5^{-9} \cdot 17^{3} \cdot 43^{-1} \cdot 339472807^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(3.9944691484360114631280273417\dots\)
Stable Faltings height: \(3.9944691484360114631280273417\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.020905792557179408696361476207\dots\)
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: \( 72 \)  = \( 1\cdot2^{3}\cdot3^{2}\cdot1 \)
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   1290.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1068480
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 1.5052170641169174261380262869325776160 \)

Local data

This elliptic curve is semistable. There are 4 primes of bad reduction:

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\((j)_{-}\)
\(2\) \(1\) \(I_{53}\) Non-split multiplicative 1 1 53 53
\(3\) \(8\) \(I_{8}\) Split multiplicative -1 1 8 8
\(5\) \(9\) \(I_{9}\) Split multiplicative -1 1 9 9
\(43\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1

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 \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 split split ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary split ordinary
$\lambda$-invariant(s) 3 3 3 0 0 0 0,0 0 0 2 0 0 0 1 0
$\mu$-invariant(s) 0 0 0 0 0 0 0,0 0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 1290.f consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$3$ 3.1.1720.1 \(\Z/2\Z\) Not in database
$6$ 6.0.5088448000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ 8.2.9690085451952.5 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.