Properties

Label 3.3.733.1-4.1-a1
Base field 3.3.733.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
 
gp: K = nfinit(Polrev([8, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-249848597a^{2}-379319816a+793737515\right){x}+2203418510276a^{2}+3345227116373a-6999982998532\)
sage: E = EllipticCurve([K([1,0,0]),K([-1,1,0]),K([1,0,0]),K([793737515,-379319816,-249848597]),K([-6999982998532,3345227116373,2203418510276])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([-1,1,0]),Polrev([1,0,0]),Polrev([793737515,-379319816,-249848597]),Polrev([-6999982998532,3345227116373,2203418510276])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![-1,1,0],K![1,0,0],K![793737515,-379319816,-249848597],K![-6999982998532,3345227116373,2203418510276]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2+a-3)\) = \((a^2+a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2+a-5)\) = \((a^2+a-3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4 \) = \(4\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2179161}{2} a^{2} - \frac{980655}{2} a + 8146597 \)
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 $\left(17822 a^{2} + 27057 a - 56618 : -5960133 a^{2} - 9048666 a + 18934591 : 1\right)$
Height \(0.16994964582045934423145139642616473783\)
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: $\left(\frac{20817}{4} a^{2} + \frac{31603}{4} a - 16533 : -\frac{20817}{8} a^{2} - \frac{31603}{8} a + 8266 : 1\right)$
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: \( 0.16994964582045934423145139642616473783 \)
Period: \( 211.00862856245316054549156674898921478 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 0.99341280734349055405285445459018112470 \)
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\((j)_{-}\)
\((a^2+a-3)\) \(4\) \(1\) \(I_{1}\) 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 4.1-a consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a \(\Q\)-curve.

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