Properties

Label 2.0.8.1-5625.2-c2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 5625 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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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(Polrev([2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}-{x}^{2}+42{x}+443\)
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([42,0]),K([443,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([42,0]),Polrev([443,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![42,0],K![443,0]]);
 

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: \((-94921875)\) = \((-a-1)^{5}\cdot(a-1)^{5}\cdot(5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9010162353515625 \) = \(3^{5}\cdot3^{5}\cdot25^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{20480}{243} \)
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 $\left(-\frac{41}{2} : \frac{275}{4} a - \frac{1}{2} : 1\right)$
Height \(1.8091757775316562435795367594540824650\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.8091757775316562435795367594540824650 \)
Period: \( 0.65492061822256392831800259490154526705 \)
Tamagawa product: \( 3 \)  =  \(1\cdot1\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 5.0269629010179055044048511259925495446 \)
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\((j)_{-}\)
\((-a-1)\) \(3\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((a-1)\) \(3\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((5)\) \(25\) \(3\) \(IV^{*}\) Additive \(1\) \(2\) \(8\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(5\) 5B.1.4

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 5625.2-c consists of curves linked by isogenies of degree 5.

Base change

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

Base field Curve
\(\Q\) 75.c2
\(\Q\) 4800.be2