Properties

Label 2.2.8.1-2450.1-e3
Base field \(\Q(\sqrt{2}) \)
Conductor norm \( 2450 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Show commands: Magma / PariGP / 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(Polrev([-2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1589a-2228\right){x}-3568a-6887\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-2228,-1589]),K([-6887,-3568])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([1,0]),Polrev([-2228,-1589]),Polrev([-6887,-3568])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-2228,-1589],K![-6887,-3568]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((35a)\) = \((a)\cdot(-2a+1)\cdot(2a+1)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2450 \) = \(2\cdot7\cdot7\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-18757812500a+95664843750)\) = \((a)^{2}\cdot(-2a+1)^{4}\cdot(2a+1)^{8}\cdot(5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -8448051270141601562500 \) = \(-2^{2}\cdot7^{4}\cdot7^{8}\cdot25^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2781996407702629}{4503750781250} a + \frac{3936073756101929}{4503750781250} \)
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: \(0\)
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: $\left(-27 a - \frac{109}{4} : \frac{27}{2} a + \frac{105}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.44344026900308057234801633515913282677 \)
Tamagawa product: \( 64 \)  =  \(2\cdot2\cdot2\cdot2^{3}\)
Torsion order: \(2\)
Leading coefficient: \( 2.5084769701061205359905008271516747502 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2a+1)\) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((5)\) \(25\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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 and 8.
Its isogeny class 2450.1-e consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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