Properties

Label 2.2.40.1-135.2-e1
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 135 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a+50\right){x}-7a+24\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,0]),K([50,-15]),K([24,-7])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,0]),Polrev([50,-15]),Polrev([24,-7])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,0],K![50,-15],K![24,-7]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a+15)\) = \((3,a+1)^{2}\cdot(3,a+2)\cdot(5,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 135 \) = \(3^{2}\cdot3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((145800a-674325)\) = \((3,a+1)^{12}\cdot(3,a+2)^{6}\cdot(5,a)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 242137805625 \) = \(3^{12}\cdot3^{6}\cdot5^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{24897088}{18225} \)
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 $\left(1 : a - 8 : 1\right)$
Height \(0.088005746879269872808975828017529058793\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1}{2} : \frac{1}{4} a - \frac{1}{2} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.088005746879269872808975828017529058793 \)
Period: \( 3.1022934405340636879494436685335007051 \)
Tamagawa product: \( 96 \)  =  \(2^{2}\cdot( 2 \cdot 3 )\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 2.0720734656171044086821131151390677875 \)
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\((j)_{-}\)
\((3,a+1)\) \(3\) \(4\) \(I_{6}^{*}\) Additive \(-1\) \(2\) \(12\) \(6\)
\((3,a+2)\) \(3\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((5,a)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 135.2-e consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.