Properties

Label 3.3.1957.1-10.1-a2
Base field 3.3.1957.1
Conductor norm \( 10 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{2}-5\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{2}-29a-30\right){x}-30a^{2}-123a-87\)
sage: E = EllipticCurve([K([-5,0,1]),K([-1,-1,0]),K([1,0,0]),K([-30,-29,-5]),K([-87,-123,-30])])
 
gp: E = ellinit([Polrev([-5,0,1]),Polrev([-1,-1,0]),Polrev([1,0,0]),Polrev([-30,-29,-5]),Polrev([-87,-123,-30])], K);
 
magma: E := EllipticCurve([K![-5,0,1],K![-1,-1,0],K![1,0,0],K![-30,-29,-5],K![-87,-123,-30]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a)\) = \((2,a)\cdot(5,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 10 \) = \(2\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((34a^2-3a-430)\) = \((2,a)^{4}\cdot(5,a)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6250000 \) = \(2^{4}\cdot5^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{15046168718351}{6250000} a^{2} + \frac{27908855989139}{6250000} a - \frac{49520546907549}{6250000} \)
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 $\left(\frac{9}{20} a^{2} - \frac{259}{100} a - \frac{63}{20} : \frac{3323}{1000} a^{2} - \frac{593}{1000} a - \frac{3021}{200} : 1\right)$
Height \(2.7552528599188446026848724257740171184\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{3}{4} a - \frac{19}{4} : \frac{11}{4} a^{2} + \frac{3}{2} a - \frac{129}{8} : 1\right)$ $\left(2 a^{2} + 3 a - 11 : -a^{2} - 5 a - 3 : 1\right)$
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: \( 2.7552528599188446026848724257740171184 \)
Period: \( 15.278839898704116921333048273906535669 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{3}\)
Torsion order: \(4\)
Leading coefficient: \( 5.7096262381248791008264610586228142042 \)
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\((j)_{-}\)
\((2,a)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((5,a)\) \(5\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 10.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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