Properties

Label 2.0.3.1-2028.2-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-52a+26)\)
Conductor norm \( 2028 \)
CM no
Base change yes: 234.c4,78.a4
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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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

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-19a+18\right){x}+704\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,0]),K([18,-19]),K([704,0])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([18,-19])),Pol(Vecrev([704,0]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,0],K![18,-19],K![704,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-52a+26)\) = \((-2a+1)\cdot(2)\cdot(-4a+1)\cdot(4a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2028 \) = \(3\cdot4\cdot13\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-207028224)\) = \((-2a+1)^{10}\cdot(2)^{16}\cdot(-4a+1)\cdot(4a-3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 42860685532594176 \) = \(3^{10}\cdot4^{16}\cdot13\cdot13\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{822656953}{207028224} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(25 a : -169 a + 84 : 1\right)$
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: \( 0.577030044571442 \)
Tamagawa product: \( 32 \)  =  \(2\cdot2^{4}\cdot1\cdot1\)
Torsion order: \(4\)
Leading coefficient: \( 1.33259380625530 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((2)\) \(4\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)
\((-4a+1)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((4a-3)\) \(13\) \(1\) \(I_{1}\) 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 and 4.
Its isogeny class 2028.2-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of elliptic curves 234.c4, 78.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.