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Properties

Label 4.4.19796.1-10.2-d1
Base field 4.4.19796.1
Conductor norm \( 10 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.19796.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + x + 8 \); class number \(1\).

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

Weierstrass equation

\({y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-7\right){x}^{2}+\left(-8a^{3}+19a^{2}+32a-58\right){x}+56a^{3}-136a^{2}-197a+316\)
sage: E = EllipticCurve([K([-3,-1,1,0]),K([-7,3,3,-1]),K([0,-5,-1,1]),K([-58,32,19,-8]),K([316,-197,-136,56])])
 
gp: E = ellinit([Polrev([-3,-1,1,0]),Polrev([-7,3,3,-1]),Polrev([0,-5,-1,1]),Polrev([-58,32,19,-8]),Polrev([316,-197,-136,56])], K);
 
magma: E := EllipticCurve([K![-3,-1,1,0],K![-7,3,3,-1],K![0,-5,-1,1],K![-58,32,19,-8],K![316,-197,-136,56]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a+2)\) = \((-a^2+2)\cdot(a^2+a-1)\)
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: \((-330a^3+679a^2+1706a-1328)\) = \((-a^2+2)^{6}\cdot(a^2+a-1)^{14}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 390625000000 \) = \(2^{6}\cdot5^{14}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{29263792417871110513}{390625000000} a^{3} - \frac{66892456727166401977}{390625000000} a^{2} - \frac{23766272438515846467}{78125000000} a + \frac{182062010179875460393}{390625000000} \)
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 $\left(-2 a + 3 : -2 a^{3} + 5 a^{2} + 6 a - 6 : 1\right)$
Height \(0.26151590357948820288690311951458405426\)
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: $\left(\frac{1}{2} a^{3} - \frac{3}{4} a^{2} - 3 a + 4 : -\frac{5}{4} a^{2} + \frac{13}{8} a + 3 : 1\right)$
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: \( 0.26151590357948820288690311951458405426 \)
Period: \( 59.104211613921247352889991596386051739 \)
Tamagawa product: \( 28 \)  =  \(2\cdot( 2 \cdot 7 )\)
Torsion order: \(2\)
Leading coefficient: \( 3.07599653732709 \)
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\((j)_{-}\)
\((-a^2+2)\) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((a^2+a-1)\) \(5\) \(14\) \(I_{14}\) Split multiplicative \(-1\) \(1\) \(14\) \(14\)

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 10.2-d 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.