Properties

Label 2.2.33.1-300.1-g6
Base field \(\Q(\sqrt{33}) \)
Conductor \((-20 a + 70)\)
Conductor norm \( 300 \)
CM no
Base change yes: 3630.w3,90.c3
Q-curve yes
Torsion order \( 12 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{33}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-8, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
 

Weierstrass equation

\(y^2+xy+\left(a+1\right)y=x^{3}+\left(-a-1\right)x^{2}+\left(122736a-413897\right)x-39525706a+133291793\)
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,1]),K([-413897,122736]),K([133291793,-39525706])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-413897,122736])),Pol(Vecrev([133291793,-39525706]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,1],K![-413897,122736],K![133291793,-39525706]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-20 a + 70)\) = \( \left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right) \cdot \left(5\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 300 \) = \( 2^{2} \cdot 3 \cdot 25 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((9000000)\) = \( \left(-a - 2\right)^{6} \cdot \left(-a + 3\right)^{6} \cdot \left(-2 a + 7\right)^{4} \cdot \left(5\right)^{6} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 81000000000000 \) = \( 2^{12} \cdot 3^{4} \cdot 25^{6} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4102915888729}{9000000} \)
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: \(\Z/2\Z\times\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-71 a + 241 : -435 a + 1464 : 1\right)$ $\left(-81 a + \frac{1099}{4} : 40 a - \frac{1103}{8} : 1\right)$
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: \( 5.36748913421879 \)
Tamagawa product: \( 864 \)  =  \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot( 2 \cdot 3 )\)
Torsion order: \(12\)
Leading coefficient: \(5.60615956111491\)
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\((j)_{-}\)
\( \left(-a - 2\right) \) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(-a + 3\right) \) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(-2 a + 7\right) \) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(5\right) \) \(25\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 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 300.1-g consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base change of elliptic curves 3630.w3, 90.c3, defined over \(\Q\), so it is also a \(\Q\)-curve.