Properties

Label 2.2.40.1-135.3-g1
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 135 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4a-5)\) = \((3,a+2)^{3}\cdot(5,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 135 \) = \(3^{3}\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-500a-625)\) = \((3,a+2)^{3}\cdot(5,a)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2109375 \) = \(3^{3}\cdot5^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{10728071}{625} a - \frac{6736561}{125} \)
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 $\left(-1 : 0 : 1\right)$
Height \(0.073508860991735727470143477354887926389\)
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: \( 0.073508860991735727470143477354887926389 \)
Period: \( 16.113616945928583209617692822213419640 \)
Tamagawa product: \( 7 \)  =  \(1\cdot7\)
Torsion order: \(1\)
Leading coefficient: \( 2.6219884172425584103183496971214167010 \)
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\((j)_{-}\)
\((3,a+2)\) \(3\) \(1\) \(II\) Additive \(1\) \(3\) \(3\) \(0\)
\((5,a)\) \(5\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 135.3-g consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

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