Properties

Label 286650.mt8
Conductor 286650
Discriminant -381473883854331756591796875000
j-invariant \( \frac{476437916651992691759}{284661685546875000} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z\)

Related objects

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, 1793932645, -5268177729853]); // or
 
magma: E := EllipticCurve("286650mt4");
 
sage: E = EllipticCurve([1, -1, 1, 1793932645, -5268177729853]) # or
 
sage: E = EllipticCurve("286650mt4")
 
gp: E = ellinit([1, -1, 1, 1793932645, -5268177729853]) \\ or
 
gp: E = ellinit("286650mt4")
 

\( y^2 + x y + y = x^{3} - x^{2} + 1793932645 x - 5268177729853 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{904101}{169}, -\frac{4657049630}{2197}\right) \)
\(\hat{h}(P)\) ≈  3.74194294656

Torsion generators

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

\( \left(\frac{11691}{4}, -\frac{11695}{8}\right) \)

Integral points

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

\( \left(752649, 653615650\right) \), \( \left(752649, -654368300\right) \)

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: \(-381473883854331756591796875000 \)  =  \(-1 \cdot 2^{3} \cdot 3^{12} \cdot 5^{18} \cdot 7^{7} \cdot 13^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{476437916651992691759}{284661685546875000} \)  =  \(2^{-3} \cdot 3^{-6} \cdot 5^{-12} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-4} \cdot 47^{3} \cdot 15107^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(3.74194294656\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0175547815323\)
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: \( 768 \)  = \( 3\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2} \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 286650.2.a.mt

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)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 339738624
\( \Gamma_0(N) \)-optimal: no
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])
 

\( L'(E,1) \) ≈ \( 12.6122862592 \)

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
\(3\) \(4\) \( I_6^{*} \) Additive -1 2 12 6
\(5\) \(4\) \( I_12^{*} \) Additive 1 2 18 12
\(7\) \(4\) \( I_1^{*} \) Additive -1 2 7 1
\(13\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4

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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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

$p$-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 non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 286650.mt 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-14}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-105}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{30}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{105}) \) \(\Z/6\Z\) Not in database
4 \(\Q(\sqrt{14}, \sqrt{30})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-14}, \sqrt{30})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
\(\Q(\sqrt{-14}, \sqrt{-30})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(i, \sqrt{105})\) \(\Z/12\Z\) Not in database
6 6.0.540027817875.2 \(\Z/6\Z\) 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.