Properties

Label 2.0.7.1-23716.5-n1
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 23716 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-7}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1045a-454\right){x}-19530a+10059\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([1,1]),K([-454,-1045]),K([10059,-19530])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([1,1]),Polrev([-454,-1045]),Polrev([10059,-19530])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![1,1],K![-454,-1045],K![10059,-19530]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((154)\) = \((a)\cdot(-a+1)\cdot(-2a+1)^{2}\cdot(-2a+3)\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23716 \) = \(2\cdot2\cdot7^{2}\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((56512151696a-44986053728)\) = \((a)^{8}\cdot(-a+1)^{4}\cdot(-2a+1)^{3}\cdot(-2a+3)^{10}\cdot(2a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5868732916160791728128 \) = \(2^{8}\cdot2^{4}\cdot7^{3}\cdot11^{10}\cdot11^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{30861084894361475}{6639980697856} a + \frac{6520918061639743}{6639980697856} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{33}{4} a - \frac{47}{2} : \frac{29}{8} a + \frac{45}{4} : 1\right)$
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: \( 0.19560816973530126264541339297633292764 \)
Tamagawa product: \( 128 \)  =  \(2^{3}\cdot2^{2}\cdot2\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 4.7317080825793671845869824408308366860 \)
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\((j)_{-}\)
\((a)\) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((-a+1)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-2a+1)\) \(7\) \(2\) \(III\) Additive \(-1\) \(2\) \(3\) \(0\)
\((-2a+3)\) \(11\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((2a+1)\) \(11\) \(1\) \(I_{5}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 23716.5-n 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.