Properties

Label 2.0.7.1-16128.5-n6
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 16128 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 8 \)
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

\({y}^2={x}^{3}+{x}^{2}-14624{x}+669300\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-14624,0]),K([669300,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-14624,0]),Polrev([669300,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-14624,0],K![669300,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-96a+48)\) = \((a)^{4}\cdot(-a+1)^{4}\cdot(-2a+1)\cdot(3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16128 \) = \(2^{4}\cdot2^{4}\cdot7\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3825245233152)\) = \((a)^{13}\cdot(-a+1)^{13}\cdot(-2a+1)^{16}\cdot(3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14632501093752098839855104 \) = \(2^{13}\cdot2^{13}\cdot7^{16}\cdot9^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{84448510979617}{933897762} \)
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 $\left(-63 a - 23 : -252 a - 1470 : 1\right)$
Height \(2.3108118147069246137026070875318320986\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(82 : 168 : 1\right)$
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: \( 2.3108118147069246137026070875318320986 \)
Period: \( 0.085636479133251818449030599440513078347 \)
Tamagawa product: \( 1024 \)  =  \(2^{2}\cdot2^{2}\cdot2^{4}\cdot2^{2}\)
Torsion order: \(8\)
Leading coefficient: \( 4.7868997978382464906921047090734783638 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{5}^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)
\((-a+1)\) \(2\) \(4\) \(I_{5}^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)
\((-2a+1)\) \(7\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)
\((3)\) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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, 4, 8 and 16.
Its isogeny class 16128.5-n consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 336.d2
\(\Q\) 2352.l2