Properties

Label 250173i1
Conductor 250173
Discriminant -24959448244269
j-invariant \( \frac{6869835701}{4991679} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 6772, -110140]) # or
 
sage: E = EllipticCurve("250173i1")
 
gp: E = ellinit([1, -1, 1, 6772, -110140]) \\ or
 
gp: E = ellinit("250173i1")
 
magma: E := EllipticCurve([1, -1, 1, 6772, -110140]); // or
 
magma: E := EllipticCurve("250173i1");
 

\( y^2 + x y + y = x^{3} - x^{2} + 6772 x - 110140 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(105, 1270\right) \)\( \left(252, 4063\right) \)
\(\hat{h}(P)\) ≈  0.79796242988391430.7853105354440074

Integral points

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

\( \left(24, 244\right) \), \( \left(24, -269\right) \), \( \left(49, 556\right) \), \( \left(49, -606\right) \), \( \left(105, 1270\right) \), \( \left(105, -1376\right) \), \( \left(225, 3460\right) \), \( \left(225, -3686\right) \), \( \left(252, 4063\right) \), \( \left(252, -4316\right) \), \( \left(1050, 33589\right) \), \( \left(1050, -34640\right) \), \( \left(82180, 23517399\right) \), \( \left(82180, -23599580\right) \), \( \left(1946764, 2715277476\right) \), \( \left(1946764, -2717224241\right) \)

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: \(-24959448244269 \)  =  \(-1 \cdot 3^{9} \cdot 7^{5} \cdot 11 \cdot 19^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{6869835701}{4991679} \)  =  \(3^{-3} \cdot 7^{-5} \cdot 11^{-1} \cdot 1901^{3}\)
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: \(0.574137273415\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.377219443728\)
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: \( 40 \)  = \( 2^{2}\cdot5\cdot1\cdot2 \)
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.i

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);
 

\( q - q^{2} - q^{4} - q^{5} + q^{7} + 3q^{8} + q^{10} + q^{11} - 4q^{13} - q^{14} - q^{16} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 460800
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 8.66302971605 \)

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\((j)_{-}\)
\(3\) \(4\) \( I_3^{*} \) Additive -1 2 9 3
\(7\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(11\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(19\) \(2\) \( III \) Additive 1 2 3 0

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(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 250173i 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 3.1.17556.1 \(\Z/2\Z\) Not in database
6 6.0.5410989815616.1 \(\Z/2\Z \times \Z/2\Z\) 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.