Properties

Label 2.0.8.1-41616.5-i2
Base field \(\Q(\sqrt{-2}) \)
Conductor \((204)\)
Conductor norm \( 41616 \)
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(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-121a-224\right){x}-1028a-811\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([-224,-121]),K([-811,-1028])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-224,-121])),Pol(Vecrev([-811,-1028]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,0],K![-224,-121],K![-811,-1028]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((204)\) = \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41616 \) = \(2^{4}\cdot3\cdot3\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((57237504a-66324480)\) = \((a)^{22}\cdot(-a-1)\cdot(a-1)^{11}\cdot(-2a+3)^{2}\cdot(2a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10951200375570432 \) = \(2^{22}\cdot3\cdot3^{11}\cdot17^{2}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2091127194353}{409563864} a + \frac{36587816460805}{1638255456} \)
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 $\left(-3 a + 6 : 37 a - 19 : 1\right)$
Height \(0.244349942130565\)
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: $\left(-3 a - 10 : 5 a - 3 : 1\right)$
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: \( 0.244349942130565 \)
Period: \( 0.545397751241716 \)
Tamagawa product: \( 88 \)  =  \(2^{2}\cdot1\cdot11\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 4.14632425398236 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{14}^{*}\) Additive \(1\) \(4\) \(22\) \(10\)
\((-a-1)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((a-1)\) \(3\) \(11\) \(I_{11}\) Split multiplicative \(-1\) \(1\) \(11\) \(11\)
\((-2a+3)\) \(17\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2a+3)\) \(17\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 41616.5-i consists of curves linked by isogenies of degree 2.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.