Properties

Label 2.0.8.1-29241.8-e1
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 29241 \)
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

\({y}^2+{y}={x}^{3}-39513{x}+3023145\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([-39513,0]),K([3023145,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-39513,0]),Polrev([3023145,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![-39513,0],K![3023145,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((171)\) = \((-a-1)^{2}\cdot(a-1)^{2}\cdot(-3a+1)\cdot(3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29241 \) = \(3^{2}\cdot3^{2}\cdot19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16245685539)\) = \((-a-1)^{8}\cdot(a-1)^{8}\cdot(-3a+1)^{5}\cdot(3a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 263922298632073720521 \) = \(3^{8}\cdot3^{8}\cdot19^{5}\cdot19^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{9358714467168256}{22284891} \)
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 $\left(-\frac{930147311}{2311250} : \frac{23897380164309}{4969187500} a - \frac{1}{2} : 1\right)$
Height \(19.803132712335879684965829395591946622\)
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: \( 19.803132712335879684965829395591946622 \)
Period: \( 0.090588610013645361239868281246685622721 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 10.148047305194686389282333090064561165 \)
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 \(-1\) \(2\) \(8\) \(2\)
\((a-1)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((-3a+1)\) \(19\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((3a+1)\) \(19\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 29241.8-e 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
\(\Q\) 171.c1
\(\Q\) 10944.bt1