Properties

Label 4.4.4525.1-45.4-c1
Base field 4.4.4525.1
Conductor norm \( 45 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.4525.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 3 x + 9 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 3, -7, -1, 1]))
 
gp: K = nfinit(Polrev([9, 3, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{7}{3}a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{4}{3}a^{2}+\frac{4}{3}a-3\right){x}^{2}+\left(-\frac{4}{3}a^{3}+\frac{1}{3}a^{2}+\frac{25}{3}a+4\right){x}-\frac{5}{3}a^{3}-\frac{4}{3}a^{2}+\frac{32}{3}a+13\)
sage: E = EllipticCurve([K([-4,-7/3,2/3,1/3]),K([-3,4/3,4/3,-1/3]),K([-4,0,1,0]),K([4,25/3,1/3,-4/3]),K([13,32/3,-4/3,-5/3])])
 
gp: E = ellinit([Polrev([-4,-7/3,2/3,1/3]),Polrev([-3,4/3,4/3,-1/3]),Polrev([-4,0,1,0]),Polrev([4,25/3,1/3,-4/3]),Polrev([13,32/3,-4/3,-5/3])], K);
 
magma: E := EllipticCurve([K![-4,-7/3,2/3,1/3],K![-3,4/3,4/3,-1/3],K![-4,0,1,0],K![4,25/3,1/3,-4/3],K![13,32/3,-4/3,-5/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^3+5/3a^2-4/3a-4)\) = \((-a^2+a+5)\cdot(1/3a^3-1/3a^2-7/3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 45 \) = \(5\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4/3a^3+22/3a^2+28/3a-7)\) = \((-a^2+a+5)^{3}\cdot(1/3a^3-1/3a^2-7/3a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 91125 \) = \(5^{3}\cdot9^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{95132237}{225} a^{3} - \frac{217836256}{135} a^{2} + \frac{232697396}{225} a + \frac{45330989}{75} \)
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 $\left(\frac{2}{3} a^{3} - \frac{2}{3} a^{2} - \frac{11}{3} a + 1 : \frac{4}{3} a^{3} - \frac{4}{3} a^{2} - \frac{22}{3} a + 2 : 1\right)$
Height \(0.0070095529150893195358878798831109860642\)
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: \( 0.0070095529150893195358878798831109860642 \)
Period: \( 1579.5321674845683160232659827920998355 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 1.97510754342395 \)
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\((j)_{-}\)
\((-a^2+a+5)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((1/3a^3-1/3a^2-7/3a+1)\) \(9\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 45.4-c consists of curves linked by isogenies of degree 3.

Base change

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

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