Properties

Label 2.2.40.1-90.1-f6
Base field \(\Q(\sqrt{10}) \)
Conductor norm \( 90 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 12 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+{y}={x}^{3}-69{x}-194\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-69,0]),K([-194,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-69,0]),Polrev([-194,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-69,0],K![-194,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a)\) = \((2,a)\cdot(3,a+1)\cdot(3,a+2)\cdot(5,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 90 \) = \(2\cdot3\cdot3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5314410)\) = \((2,a)^{2}\cdot(3,a+1)^{12}\cdot(3,a+2)^{12}\cdot(5,a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 28242953648100 \) = \(2^{2}\cdot3^{12}\cdot3^{12}\cdot5^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{35578826569}{5314410} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-2 a + 3 : a - 2 : 1\right)$ $\left(-4 : 6 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.8088892831580245719059150358786380324 \)
Tamagawa product: \( 576 \)  =  \(2\cdot( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot2\)
Torsion order: \(12\)
Leading coefficient: \( 1.7764975660033988509252275712977984037 \)
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\((j)_{-}\)
\((2,a)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((3,a+1)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((3,a+2)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((5,a)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 90.1-f consists of curves linked by isogenies of degrees dividing 24.

Base change

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

Base field Curve
\(\Q\) 30.a5
\(\Q\) 4800.cq5