Properties

Label 3.3.1620.1-25.1-a2
Base field 3.3.1620.1
Conductor norm \( 25 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 6 \)
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

\({y}^2+{y}={x}^{3}+\left(a^{2}-2a-9\right){x}^{2}+\left(527694250a^{2}-733973258a-5311442969\right){x}-11070357693729a^{2}+15397830281951a+111427353147970\)
sage: E = EllipticCurve([K([0,0,0]),K([-9,-2,1]),K([1,0,0]),K([-5311442969,-733973258,527694250]),K([111427353147970,15397830281951,-11070357693729])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([-9,-2,1])),Pol(Vecrev([1,0,0])),Pol(Vecrev([-5311442969,-733973258,527694250])),Pol(Vecrev([111427353147970,15397830281951,-11070357693729]))], K);
 
magma: E := EllipticCurve([K![0,0,0],K![-9,-2,1],K![1,0,0],K![-5311442969,-733973258,527694250],K![111427353147970,15397830281951,-11070357693729]]);
 

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: \((125)\) = \((a+3)^{6}\cdot(2a^2-3a-19)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1953125 \) = \(5^{6}\cdot5^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2359296}{125} \)
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 $\left(-\frac{158076}{25} a^{2} + \frac{219874}{25} a + \frac{1591086}{25} : -\frac{15917422}{125} a^{2} + \frac{22139653}{125} a + \frac{160214792}{125} : 1\right)$
Height \(0.528563194265773\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2950 a^{2} + 4103 a + 29694 : -623624 a^{2} + 867402 a + 6277014 : 1\right)$
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: \( 0.528563194265773 \)
Period: \( 194.308052142908 \)
Tamagawa product: \( 18 \)  =  \(( 2 \cdot 3 )\cdot3\)
Torsion order: \(6\)
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\((j)_{-}\)
\((a+3)\) \(5\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((2a^2-3a-19)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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.1.1

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.d2