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Properties

Label 4.4.13025.1-16.1-d1
Base field 4.4.13025.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.13025.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 3 x + 29 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 3, -12, -1, 1]))
 
gp: K = nfinit(Polrev([29, 3, -12, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 3, -12, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{2}-6\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{5}{4}\right){x}^{2}+\left(\frac{143}{2}a^{3}-246a^{2}-261a+\frac{1709}{2}\right){x}+\frac{3943}{4}a^{3}-\frac{6755}{2}a^{2}-\frac{7279}{2}a+\frac{47147}{4}\)
sage: E = EllipticCurve([K([1,0,0,0]),K([-5/4,1/2,1/2,-1/4]),K([-6,0,1,0]),K([1709/2,-261,-246,143/2]),K([47147/4,-7279/2,-6755/2,3943/4])])
 
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-5/4,1/2,1/2,-1/4]),Polrev([-6,0,1,0]),Polrev([1709/2,-261,-246,143/2]),Polrev([47147/4,-7279/2,-6755/2,3943/4])], K);
 
magma: E := EllipticCurve([K![1,0,0,0],K![-5/4,1/2,1/2,-1/4],K![-6,0,1,0],K![1709/2,-261,-246,143/2],K![47147/4,-7279/2,-6755/2,3943/4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((-3/4a^3-3/2a^2+5/2a+21/4)\cdot(1/2a^3-2a^2-3a+27/2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(4\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^2-4a-6)\) = \((-3/4a^3-3/2a^2+5/2a+21/4)\cdot(1/2a^3-2a^2-3a+27/2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4096 \) = \(4\cdot4^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{181766666214515}{64} a^{3} + \frac{113866544096805}{16} a^{2} - \frac{146107414893665}{16} a - \frac{1503577557583359}{64} \)
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: \( 21.123758696394063656924337857645076882 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 0.185089734208271 \)
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\((j)_{-}\)
\((-3/4a^3-3/2a^2+5/2a+21/4)\) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((1/2a^3-2a^2-3a+27/2)\) \(4\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 16.1-d consists of curves linked by isogenies of degree 5.

Base change

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

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