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Properties

Label 5.5.24217.1-29.1-b1
Base field 5.5.24217.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.24217.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
 
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-4a^{2}\right){x}{y}+\left(2a^{4}-9a^{2}+2\right){y}={x}^{3}+\left(-3a^{4}+14a^{2}+a-4\right){x}^{2}+\left(262a^{4}-221a^{3}-1174a^{2}+730a+366\right){x}+3311a^{4}-2134a^{3}-14758a^{2}+6316a+3956\)
sage: E = EllipticCurve([K([0,0,-4,0,1]),K([-4,1,14,0,-3]),K([2,0,-9,0,2]),K([366,730,-1174,-221,262]),K([3956,6316,-14758,-2134,3311])])
 
gp: E = ellinit([Polrev([0,0,-4,0,1]),Polrev([-4,1,14,0,-3]),Polrev([2,0,-9,0,2]),Polrev([366,730,-1174,-221,262]),Polrev([3956,6316,-14758,-2134,3311])], K);
 
magma: E := EllipticCurve([K![0,0,-4,0,1],K![-4,1,14,0,-3],K![2,0,-9,0,2],K![366,730,-1174,-221,262],K![3956,6316,-14758,-2134,3311]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^4+a^3+10a^2-2a-4)\) = \((-2a^4+a^3+10a^2-2a-4)\)
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: \((204a^4-90a^3-966a^2+167a+465)\) = \((-2a^4+a^3+10a^2-2a-4)^{7}\)
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: \( -\frac{1081740781776866294940863471759}{17249876309} a^{4} + \frac{2120240724895871393224432246770}{17249876309} a^{3} + \frac{43206056991447348862819034032}{594823321} a^{2} - \frac{1374124848200655188432751002870}{17249876309} a - \frac{551901067153896026723504870757}{17249876309} \)
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: \( 0.43696576436689693633201083830091896311 \)
Tamagawa product: \( 7 \)
Torsion order: \(1\)
Leading coefficient: \( 0.963122128 \)
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\((j)_{-}\)
\((-2a^4+a^3+10a^2-2a-4)\) \(29\) \(7\) \(I_{7}\) 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-b 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.