Properties

Label 2.2.12.1-75.1-b7
Base field \(\Q(\sqrt{3}) \)
Conductor \((5a)\)
Conductor norm \( 75 \)
CM no
Base change yes: 15.a4,720.c4
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-3, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4481a-7761\right){x}-217866a+377355\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([-7761,4481]),K([377355,-217866])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-7761,4481])),Pol(Vecrev([377355,-217866]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,0],K![-7761,4481],K![377355,-217866]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((5a)\) = \((a)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 75 \) = \(3\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((15)\) = \((a)^{2}\cdot(5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 225 \) = \(3^{2}\cdot25\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{56667352321}{15} \)
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 $\left(-24 a + \frac{83}{2} : \frac{99}{4} a - \frac{171}{4} : 1\right)$
Height \(1.30031284399313\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-21 a + 36 : -17 a + 30 : 1\right)$
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: \( 1.30031284399313 \)
Period: \( 31.3870221150614 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(8\)
Leading coefficient: \( 0.736355203405453 \)
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\((j)_{-}\)
\((a)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((5)\) \(25\) \(1\) \(I_{1}\) 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, 4, 8 and 16.
Its isogeny class 75.1-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of elliptic curves 15.a4, 720.c4, defined over \(\Q\), so it is also a \(\Q\)-curve.