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Properties

Label 5.5.70601.1-29.1-a1
Base field 5.5.70601.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.70601.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-6a^{2}-a+3\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+a\right){y}={x}^{3}+\left(-a^{4}+7a^{2}+a-6\right){x}^{2}+\left(27a^{4}+49a^{3}-206a^{2}-213a-28\right){x}+2022a^{4}+73a^{3}-11190a^{2}-5360a+2507\)
sage: E = EllipticCurve([K([3,-1,-6,0,1]),K([-6,1,7,0,-1]),K([0,1,-4,-1,1]),K([-28,-213,-206,49,27]),K([2507,-5360,-11190,73,2022])])
 
gp: E = ellinit([Polrev([3,-1,-6,0,1]),Polrev([-6,1,7,0,-1]),Polrev([0,1,-4,-1,1]),Polrev([-28,-213,-206,49,27]),Polrev([2507,-5360,-11190,73,2022])], K);
 
magma: E := EllipticCurve([K![3,-1,-6,0,1],K![-6,1,7,0,-1],K![0,1,-4,-1,1],K![-28,-213,-206,49,27],K![2507,-5360,-11190,73,2022]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^4-2a^3-9a^2+2a+3)\) = \((2a^4-2a^3-9a^2+2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-73a^4+55a^3+396a^2+3a-137)\) = \((2a^4-2a^3-9a^2+2a+3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 17249876309 \) = \(29^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1499741787190786782165925}{17249876309} a^{4} - \frac{4214003745944059872006099}{17249876309} a^{3} + \frac{74873005092991374350620}{17249876309} a^{2} + \frac{2994939658199456178076242}{17249876309} a - \frac{879336253518581869182340}{17249876309} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.61069271502342918430240996389981894683 \)
Tamagawa product: \( 7 \)
Torsion order: \(1\)
Leading coefficient: \( 0.788336134 \)
Analytic order of Ш: \( 49 \) (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\((j)_{-}\)
\((2a^4-2a^3-9a^2+2a+3)\) \(29\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 29.1-a consists of curves linked by isogenies of degree 7.

Base change

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

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