Properties

Label 2.0.7.1-28672.7-i2
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 28672 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2={x}^{3}-a{x}^{2}+\left(-2a+1\right){x}+a-4\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([1,-2]),K([-4,1])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([1,-2]),Polrev([-4,1])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![1,-2],K![-4,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-128a+64)\) = \((a)^{6}\cdot(-a+1)^{6}\cdot(-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28672 \) = \(2^{6}\cdot2^{6}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((64a-704)\) = \((a)^{6}\cdot(-a+1)^{10}\cdot(-2a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 458752 \) = \(2^{6}\cdot2^{10}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{21696}{7} a - \frac{30784}{7} \)
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 $\left(\frac{1}{4} a + \frac{1}{2} : \frac{9}{8} a - \frac{3}{4} : 1\right)$
Height \(1.3334890985721596623501007040326029208\)
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: $\left(a : 0 : 1\right)$
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: \( 1.3334890985721596623501007040326029208 \)
Period: \( 4.3501445127523126301455778738936618979 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 4.3850457605061670398646857234168911877 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(II\) Additive \(1\) \(6\) \(6\) \(0\)
\((-a+1)\) \(2\) \(2\) \(I_0^{*}\) Additive \(1\) \(6\) \(10\) \(0\)
\((-2a+1)\) \(7\) \(1\) \(I_{1}\) 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 28672.7-i consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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