Properties

Label 102550z1
Conductor 102550
Discriminant -13461123200000000
j-invariant \( -\frac{12274557745}{34460475392} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -3513, -5582983]); // or
 
magma: E := EllipticCurve("102550z1");
 
sage: E = EllipticCurve([1, 0, 0, -3513, -5582983]) # or
 
sage: E = EllipticCurve("102550z1")
 
gp: E = ellinit([1, 0, 0, -3513, -5582983]) \\ or
 
gp: E = ellinit("102550z1")
 

\( y^2 + x y = x^{3} - 3513 x - 5582983 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(202, 1299\right) \)\( \left(286, 3959\right) \)
\(\hat{h}(P)\) ≈  0.7613190583821.03147589388

Integral points

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

\( \left(202, 1299\right) \), \( \left(202, -1501\right) \), \( \left(286, 3959\right) \), \( \left(286, -4245\right) \), \( \left(298, 4307\right) \), \( \left(298, -4605\right) \), \( \left(952, 28749\right) \), \( \left(952, -29701\right) \), \( \left(1402, 51699\right) \), \( \left(1402, -53101\right) \), \( \left(3802, 232499\right) \), \( \left(3802, -236301\right) \), \( \left(9076, 860105\right) \), \( \left(9076, -869181\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 102550 \)  =  \(2 \cdot 5^{2} \cdot 7 \cdot 293\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-13461123200000000 \)  =  \(-1 \cdot 2^{13} \cdot 5^{8} \cdot 7^{2} \cdot 293^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{12274557745}{34460475392} \)  =  \(-1 \cdot 2^{-13} \cdot 5 \cdot 7^{-2} \cdot 19^{3} \cdot 71^{3} \cdot 293^{-2}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.646189887682\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.180259990315\)
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: \( 156 \)  = \( 13\cdot3\cdot2\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 102550.2.a.w

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)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 836160
\( \Gamma_0(N) \)-optimal: yes
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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 18.1712205317 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(13\) \( I_{13} \) Split multiplicative -1 1 13 13
\(5\) \(3\) \( IV^{*} \) Additive -1 2 8 0
\(7\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(293\) \(2\) \( I_{2} \) 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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 \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 293
Reduction type split ordinary add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary split
$\lambda$-invariant(s) 5 2 - 2 2 2 2 2 2 2 2 2 2 2,4 2 3
$\mu$-invariant(s) 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 no rational isogenies. Its isogeny class 102550z consists of this curve only.

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 3.1.200.1 \(\Z/2\Z\) Not in database
6 6.0.320000.1 \(\Z/2\Z \times \Z/2\Z\) 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.