Properties

Label 2.2.33.1-33.1-a6
Base field \(\Q(\sqrt{33}) \)
Conductor norm \( 33 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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(Polrev([-8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-944a-2235\right){x}-27553a-65366\)
sage: E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([-2235,-944]),K([-65366,-27553])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([-2235,-944]),Polrev([-65366,-27553])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![-2235,-944],K![-65366,-27553]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+1)\) = \((-2a+7)\cdot(-4a-9)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 33 \) = \(3\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((6a-3)\) = \((-2a+7)^{3}\cdot(-4a-9)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -297 \) = \(-3^{3}\cdot11\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{181064514814}{99} a + \frac{429535966213}{99} \)
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 $\left(-\frac{87317}{11264} a - \frac{24269}{1408} : \frac{13326785}{3964928} a + \frac{4242489}{495616} : 1\right)$
Height \(5.9154020089622002142454346120997449347\)
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: $\left(-\frac{31}{4} a - \frac{69}{4} : \frac{27}{8} a + \frac{69}{8} : 1\right)$
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: \( 5.9154020089622002142454346120997449347 \)
Period: \( 4.4681264139908030044721994486572592216 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 2.3005027181265763143156656624447302351 \)
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\((j)_{-}\)
\((-2a+7)\) \(3\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-4a-9)\) \(11\) \(1\) \(I_{1}\) 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 and 8.
Its isogeny class 33.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.