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Properties

Label 4.4.13025.1-16.1-a2
Base field 4.4.13025.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+2a^{2}+3a-\frac{15}{2}\right){x}^{2}+\left(-18a^{3}-36a^{2}+60a+121\right){x}+142a^{3}+366a^{2}-457a-1212\)
sage: E = EllipticCurve([K([0,1,0,0]),K([-15/2,3,2,-1/2]),K([-6,-1,1,0]),K([121,60,-36,-18]),K([-1212,-457,366,142])])
 
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-15/2,3,2,-1/2]),Polrev([-6,-1,1,0]),Polrev([121,60,-36,-18]),Polrev([-1212,-457,366,142])], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![-15/2,3,2,-1/2],K![-6,-1,1,0],K![121,60,-36,-18],K![-1212,-457,366,142]]);
 

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: \((32)\) = \((-3/4a^3-3/2a^2+5/2a+21/4)^{5}\cdot(1/2a^3-2a^2-3a+27/2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1048576 \) = \(4^{5}\cdot4^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{787971}{64} a^{3} + \frac{874133}{32} a^{2} - \frac{734245}{16} a - \frac{6087581}{64} \)
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: \(1\)
Generator $\left(a^{2} - 3 : -2 a^{3} + a^{2} + 7 a - 3 : 1\right)$
Height \(0.058780720992618310440920553682224554688\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.058780720992618310440920553682224554688 \)
Period: \( 269.23321316713301836253489734333844019 \)
Tamagawa product: \( 5 \)  =  \(1\cdot5\)
Torsion order: \(1\)
Leading coefficient: \( 2.77334994402905 \)
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\((j)_{-}\)
\((-3/4a^3-3/2a^2+5/2a+21/4)\) \(4\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((1/2a^3-2a^2-3a+27/2)\) \(4\) \(5\) \(I_{5}\) 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-a 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.