Properties

Label 286650.t1
Conductor 286650
Discriminant 23896482544279800000000
j-invariant \( \frac{1551349793665}{14556672} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -28450242, -57926063084]); // or
 
magma: E := EllipticCurve("286650t1");
 
sage: E = EllipticCurve([1, -1, 0, -28450242, -57926063084]) # or
 
sage: E = EllipticCurve("286650t1")
 
gp: E = ellinit([1, -1, 0, -28450242, -57926063084]) \\ or
 
gp: E = ellinit("286650t1")
 

\( y^2 + x y = x^{3} - x^{2} - 28450242 x - 57926063084 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-3295, 7601\right) \)\( \left(20519, 2817653\right) \)
\(\hat{h}(P)\) ≈  2.020008984372.30708016309

Integral points

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

\( \left(-3295, 7601\right) \), \( \left(-3295, -4306\right) \), \( \left(-3181, 21053\right) \), \( \left(-3181, -17872\right) \), \( \left(20519, 2817653\right) \), \( \left(20519, -2838172\right) \), \( \left(126653, 44969609\right) \), \( \left(126653, -45096262\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 286650 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(23896482544279800000000 \)  =  \(2^{9} \cdot 3^{13} \cdot 5^{8} \cdot 7^{8} \cdot 13 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{1551349793665}{14556672} \)  =  \(2^{-9} \cdot 3^{-7} \cdot 5 \cdot 7 \cdot 13^{-1} \cdot 3539^{3}\)
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: \(4.65948093262\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0654069703685\)
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: \( 36 \)  = \( 1\cdot2^{2}\cdot3\cdot3\cdot1 \)
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 286650.2.a.t

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)
 

\( q - q^{2} + q^{4} - q^{8} - 5q^{11} + q^{13} + q^{16} + 8q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 35562240
\( \Gamma_0(N) \)-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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 10.9714511265 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{9} \) Non-split multiplicative 1 1 9 9
\(3\) \(4\) \( I_7^{*} \) Additive -1 2 13 7
\(5\) \(3\) \( IV^{*} \) Additive -1 2 8 0
\(7\) \(3\) \( IV^{*} \) Additive 1 2 8 0
\(13\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 286650.t 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 3.3.382200.1 \(\Z/2\Z\) Not in database
6 6.6.45575974080000.1 \(\Z/2\Z \times \Z/2\Z\) 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.