Properties

Label 3.3.1957.1-2.1-a5
Base field 3.3.1957.1
Conductor norm \( 2 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

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Base field 3.3.1957.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, -9, -1, 1]))
 
gp: K = nfinit(Polrev([10, -9, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(-26a^{2}+100a-77\right){x}-198a^{2}+767a-549\)
sage: E = EllipticCurve([K([1,0,0]),K([5,1,-1]),K([1,0,0]),K([-77,100,-26]),K([-549,767,-198])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([5,1,-1]),Polrev([1,0,0]),Polrev([-77,100,-26]),Polrev([-549,767,-198])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![5,1,-1],K![1,0,0],K![-77,100,-26],K![-549,767,-198]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2,a)\) = \((2,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2 \) = \(2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3a^2+5a-18)\) = \((2,a)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1024 \) = \(-2^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22236981725625867673}{1024} a^{2} + \frac{42506064262246252101}{1024} a - \frac{76376288557509693211}{1024} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(3 a^{2} - 12 a + 11 : 23 a^{2} - 88 a + 60 : 1\right)$
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: \( 39.924446262993718961837100267147498650 \)
Tamagawa product: \( 10 \)
Torsion order: \(4\)
Leading coefficient: \( 2.2562306632343262932577151134648946801 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((2,a)\) \(2\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

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
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 10 and 20.
Its isogeny class 2.1-a consists of curves linked by isogenies of degrees dividing 20.

Base change

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

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