Properties

Label 2.0.8.1-18513.14-d1
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 18513 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-7a-21\right){x}+53a+25\)
sage: E = EllipticCurve([K([0,1]),K([0,-1]),K([1,1]),K([-21,-7]),K([25,53])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([-21,-7]),Polrev([25,53])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,-1],K![1,1],K![-21,-7],K![25,53]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-26a+131)\) = \((a-1)^{2}\cdot(a+3)^{2}\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 18513 \) = \(3^{2}\cdot11^{2}\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-439438a+1724209)\) = \((a-1)^{8}\cdot(a+3)^{6}\cdot(2a+3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3359108187369 \) = \(3^{8}\cdot11^{6}\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1437952}{2601} a - \frac{95168}{2601} \)
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 $\left(-\frac{8}{9} a + \frac{67}{72} : \frac{3707}{864} a + \frac{545}{108} : 1\right)$
Height \(3.3659219589064621075633199885215814295\)
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: $\left(2 a + \frac{3}{2} : -\frac{5}{4} a + \frac{3}{2} : 1\right)$
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: \( 3.3659219589064621075633199885215814295 \)
Period: \( 1.2398526677229902387451377452172575830 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 11.803725918438313615518086081234345547 \)
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\((j)_{-}\)
\((a-1)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((a+3)\) \(11\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((2a+3)\) \(17\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 18513.14-d consists of curves linked by isogenies of degree 2.

Base change

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

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