Properties

Label 3.3.1620.1-25.1-a3
Base field 3.3.1620.1
Conductor norm \( 25 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Base field 3.3.1620.1

Generator \(a\), with minimal polynomial \( x^{3} - 12 x - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -12, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-14, -12, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -12, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-7\right){y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(2053627a^{2}-2856402a-20670534\right){x}+2802623882a^{2}-3898187220a-28209473416\)
sage: E = EllipticCurve([K([0,0,0]),K([-8,-1,1]),K([-7,-1,1]),K([-20670534,-2856402,2053627]),K([-28209473416,-3898187220,2802623882])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([-8,-1,1])),Pol(Vecrev([-7,-1,1])),Pol(Vecrev([-20670534,-2856402,2053627])),Pol(Vecrev([-28209473416,-3898187220,2802623882]))], K);
 
magma: E := EllipticCurve([K![0,0,0],K![-8,-1,1],K![-7,-1,1],K![-20670534,-2856402,2053627],K![-28209473416,-3898187220,2802623882]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-3a-3)\) = \((a+3)\cdot(2a^2-3a-19)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \(5\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5)\) = \((a+3)^{2}\cdot(2a^2-3a-19)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 125 \) = \(5^{2}\cdot5\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{884736}{5} \)
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 $\left(-\frac{82199}{100} a^{2} + \frac{57163}{50} a + \frac{206816}{25} : \frac{4500237}{125} a^{2} - \frac{25037727}{500} a - \frac{181187153}{500} : 1\right)$
Height \(1.58568958279732\)
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: $\left(351 a^{2} - \frac{977}{2} a - 3533 : -\frac{1}{2} a^{2} + \frac{1}{2} a + \frac{7}{2} : 1\right)$
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: \( 1.58568958279732 \)
Period: \( 64.7693507143025 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.82755524972352 \)
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\((j)_{-}\)
\((a+3)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2a^2-3a-19)\) \(5\) \(1\) \(I_{1}\) 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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 25.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q\) 405.d1