Properties

Label 2.0.3.1-28812.3-f1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-196a+98)\)
Conductor norm \( 28812 \)
CM no
Base change yes: 294.g5,882.b5
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-197a+197\right){x}-2367\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([197,-197]),K([-2367,0])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([197,-197])),Pol(Vecrev([-2367,0]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![197,-197],K![-2367,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-196a+98)\) = \((-2a+1)\cdot(2)\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28812 \) = \(3\cdot4\cdot7^{2}\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1897443072)\) = \((-2a+1)^{4}\cdot(2)^{8}\cdot(-3a+1)^{7}\cdot(3a-2)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3600290211480797184 \) = \(3^{4}\cdot4^{8}\cdot7^{7}\cdot7^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7189057}{16128} \)
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 $\left(-2 a + 8 : -54 a + 39 : 1\right)$
Height \(0.138307751467270\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-46 a : 46 a - 317 : 1\right)$
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: \( 0.138307751467270 \)
Period: \( 0.391481047466294 \)
Tamagawa product: \( 512 \)  =  \(2^{2}\cdot2^{3}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 4.00135058804565 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((2)\) \(4\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((-3a+1)\) \(7\) \(4\) \(I_1^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((3a-2)\) \(7\) \(4\) \(I_1^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)

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, 4 and 8.
Its isogeny class 28812.3-f consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 294.g5, 882.b5, defined over \(\Q\), so it is also a \(\Q\)-curve.