Properties

Label 2.2.56.1-10.2-c2
Base field \(\Q(\sqrt{14}) \)
Conductor norm \( 10 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{14}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
 
gp: K = nfinit(Polrev([-14, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(5a+29\right){x}+20a+81\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([29,5]),K([81,20])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([29,5]),Polrev([81,20])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![29,5],K![81,20]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a+2)\) = \((-a+4)\cdot(-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 10 \) = \(2\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((928a-2976)\) = \((-a+4)^{10}\cdot(-a-3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3200000 \) = \(-2^{10}\cdot5^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{73603923}{100000} a + \frac{358833109}{100000} \)
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 $\left(-2 : -3 : 1\right)$
Height \(0.054400494674780956742708434690663425336\)
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: \( 0.054400494674780956742708434690663425336 \)
Period: \( 14.163134738548721329178436662472233718 \)
Tamagawa product: \( 10 \)  =  \(( 2 \cdot 5 )\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.0591985216127295013337256651752767828 \)
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\((j)_{-}\)
\((-a+4)\) \(2\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((-a-3)\) \(5\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 10.2-c consists of curves linked by isogenies of degree 5.

Base change

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

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