Properties

Label 2.2.40.1-245.1-a1
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 245 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{10}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, 0, 1]))
 
gp: K = nfinit(Polrev([-10, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+{x}^{2}-131{x}-650\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([-131,0]),K([-650,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([1,0]),Polrev([-131,0]),Polrev([-650,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![-131,0],K![-650,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((35,7a)\) = \((5,a)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 245 \) = \(5\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-13671875)\) = \((5,a)^{18}\cdot(7)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 186920166015625 \) = \(5^{18}\cdot49\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{250523582464}{13671875} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{168677}{3042} : \frac{30538399}{237276} a - \frac{1}{2} : 1\right)$
Height \(12.154716343954060768311603268122231553\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 12.154716343954060768311603268122231553 \)
Period: \( 0.49408421010216832663347983667534091830 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 3.7981822402645628065163144682199668938 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((5,a)\) \(5\) \(2\) \(I_{18}\) Non-split multiplicative \(1\) \(1\) \(18\) \(18\)
\((7)\) \(49\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 245.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 35.a1
\(\Q\) 11200.cg1