Properties

Label 2.0.7.1-15488.5-f4
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 15488 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(20a-42\right){x}-68a+79\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-42,20]),K([79,-68])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-42,20]),Polrev([79,-68])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-42,20],K![79,-68]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-22a+132)\) = \((a)^{6}\cdot(-a+1)\cdot(-2a+3)\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 15488 \) = \(2^{6}\cdot2\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-15664a+32032)\) = \((a)^{19}\cdot(-a+1)^{4}\cdot(-2a+3)\cdot(2a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1015021568 \) = \(2^{19}\cdot2^{4}\cdot11\cdot11\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{34643161}{176} a + \frac{13683239}{88} \)
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 $\left(-\frac{67}{64} a + \frac{121}{32} : -\frac{753}{512} a - \frac{349}{256} : 1\right)$
Height \(2.4183391352709074809368818561627269463\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 2 : -7 : 1\right)$
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: \( 2.4183391352709074809368818561627269463 \)
Period: \( 1.7379587607853794823609330650473907946 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\cdot1\cdot1\)
Torsion order: \(4\)
Leading coefficient: \( 6.3542989382521078342040216286068828413 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{9}^{*}\) Additive \(-1\) \(6\) \(19\) \(1\)
\((-a+1)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-2a+3)\) \(11\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2a+1)\) \(11\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 and 4.
Its isogeny class 15488.5-f consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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