Properties

Label 2.0.7.1-14400.1-d6
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 14400 \)
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

\({y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-675a-271\right){x}-10542a+4229\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,0]),K([-271,-675]),K([4229,-10542])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-271,-675]),Polrev([4229,-10542])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,0],K![-271,-675],K![4229,-10542]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((75a+30)\) = \((a)^{6}\cdot(3)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 14400 \) = \(2^{6}\cdot9\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((13942125a-88901550)\) = \((a)^{18}\cdot(3)^{8}\cdot(5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7052774768640000 \) = \(2^{18}\cdot9^{8}\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{272223782641}{164025} \)
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 $\left(\frac{12091}{3364} a + \frac{11231}{1682} : -\frac{17666943}{195112} a + \frac{14617761}{97556} : 1\right)$
Height \(7.6380040489745907919641634275790386194\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{29}{4} a - \frac{27}{2} : \frac{83}{8} a - \frac{29}{4} : 1\right)$ $\left(-7 a - 13 : 10 a - 7 : 1\right)$
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: \( 7.6380040489745907919641634275790386194 \)
Period: \( 0.39521996042555817677790773532444837400 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 4.5638328067218646116148836222873291942 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{8}^{*}\) Additive \(1\) \(6\) \(18\) \(0\)
\((3)\) \(9\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((5)\) \(25\) \(2\) \(I_{2}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 14400.1-d consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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