Properties

Label 2.0.8.1-5625.2-b5
Base field \(\Q(\sqrt{-2}) \)
Conductor \((75)\)
Conductor norm \( 5625 \)
CM no
Base change yes: 4800.bz6,75.b6
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-2}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-126{x}+523\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-126,0]),K([523,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-126,0])),Pol(Vecrev([523,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-126,0],K![523,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((75)\) = \((-a-1)\cdot(a-1)\cdot(5)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5625 \) = \(3\cdot3\cdot25^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3515625)\) = \((-a-1)^{2}\cdot(a-1)^{2}\cdot(5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 12359619140625 \) = \(3^{2}\cdot3^{2}\cdot25^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{13997521}{225} \)
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 $\left(-\frac{593}{36} : \frac{1625}{54} a + \frac{557}{72} : 1\right)$
Height \(4.29778374490333\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-13 : 6 : 1\right)$ $\left(\frac{23}{4} : -\frac{27}{8} : 1\right)$
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: \( 4.29778374490333 \)
Period: \( 0.894280685047011 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.43542374905056 \)
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\((j)_{-}\)
\((-a-1)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a-1)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((5)\) \(25\) \(4\) \(I_2^{*}\) Additive \(1\) \(2\) \(8\) \(2\)

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 5625.2-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of 4800.bz6, 75.b6, defined over \(\Q\), so it is also a \(\Q\)-curve.