Properties

Label 250173.b1
Conductor 250173
Discriminant 287210199947173496599221
j-invariant \( \frac{15985030403346927616}{8374342621029} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -170508603, -856585839470]) # or
 
sage: E = EllipticCurve("250173b2")
 
gp: E = ellinit([0, 0, 1, -170508603, -856585839470]) \\ or
 
gp: E = ellinit("250173b2")
 
magma: E := EllipticCurve([0, 0, 1, -170508603, -856585839470]); // or
 
magma: E := EllipticCurve("250173b2");
 

\( y^2 + y = x^{3} - 170508603 x - 856585839470 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-\frac{2474183287}{322624}, -\frac{357706393563}{183250432}\right) \)\( \left(\frac{67393}{4}, -\frac{8210227}{8}\right) \)
\(\hat{h}(P)\) ≈  13.8594213326072053.4898256204087077

Integral points

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

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 250173 \)  =  \(3^{2} \cdot 7 \cdot 11 \cdot 19^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(287210199947173496599221 \)  =  \(3^{7} \cdot 7 \cdot 11^{5} \cdot 19^{11} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{15985030403346927616}{8374342621029} \)  =  \(2^{12} \cdot 3^{-1} \cdot 7^{-1} \cdot 11^{-5} \cdot 19^{-5} \cdot 29^{3} \cdot 61^{3} \cdot 89^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(9.39386109208\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.041780788939\)
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: \( 16 \)  = \( 2^{2}\cdot1\cdot1\cdot2^{2} \)
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: \(1\) (rounded)

Modular invariants

Modular form 250173.2.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);
 

\( q - 2q^{2} + 2q^{4} - q^{5} + q^{7} + 2q^{10} - q^{11} - 4q^{13} - 2q^{14} - 4q^{16} - 8q^{17} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 74880000
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 6.27972684177 \)

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\((j)_{-}\)
\(3\) \(4\) \( I_1^{*} \) Additive -1 2 7 1
\(7\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(11\) \(1\) \( I_{5} \) Non-split multiplicative 1 1 5 5
\(19\) \(4\) \( I_5^{*} \) Additive -1 2 11 5

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.4.2

$p$-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 \(d=\) 5.
Its isogeny class 250173.b 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 3.3.17556.1 \(\Z/2\Z\) Not in database
4 4.0.406125.2 \(\Z/5\Z\) Not in database
6 6.6.1352747453904.1 \(\Z/2\Z \times \Z/2\Z\) 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.