Properties

Label 3.3.1620.1-25.1-b2
Base field 3.3.1620.1
Conductor \((a^2-3a-3)\)
Conductor norm \( 25 \)
CM no
Base change yes: 405.a1
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+9\right){x}^{2}+\left(20a^{2}-28a-200\right){x}+108a^{2}-151a-1090\)
sage: E = EllipticCurve([K([0,0,0]),K([9,2,-1]),K([1,1,0]),K([-200,-28,20]),K([-1090,-151,108])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([9,2,-1])),Pol(Vecrev([1,1,0])),Pol(Vecrev([-200,-28,20])),Pol(Vecrev([-1090,-151,108]))], K);
 
magma: E := EllipticCurve([K![0,0,0],K![9,2,-1],K![1,1,0],K![-200,-28,20],K![-1090,-151,108]]);
 

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: \( \frac{36864}{5} \)
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 $\left(-4 a^{2} + 5 a + 42 : -23 a^{2} + 32 a + 229 : 1\right)$
Height \(0.272277013504516\)
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: $\left(a^{2} - \frac{3}{2} a - 10 : -\frac{1}{2} a - \frac{1}{2} : 1\right)$
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: \( 0.272277013504516 \)
Period: \( 147.106329377527 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.49271222347893 \)
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\((j)_{-}\)
\((a+3)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2a^2-3a-19)\) \(5\) \(1\) \(I_{1}\) 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 25.1-b consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of 405.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.