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Properties

Label 4.4.5725.1-9.1-a2
Base field 4.4.5725.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.5725.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 6, -8, -1, 1]))
 
gp: K = nfinit(Polrev([11, 6, -8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 6, -8, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{3}a^{3}-\frac{5}{3}a+\frac{4}{3}\right){x}{y}+\left(\frac{1}{3}a^{3}-\frac{5}{3}a+\frac{4}{3}\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-\frac{20}{3}a^{3}-2a^{2}+\frac{124}{3}a+\frac{49}{3}\right){x}+\frac{25}{3}a^{3}+2a^{2}-\frac{170}{3}a-\frac{74}{3}\)
sage: E = EllipticCurve([K([4/3,-5/3,0,1/3]),K([-3,0,1,0]),K([4/3,-5/3,0,1/3]),K([49/3,124/3,-2,-20/3]),K([-74/3,-170/3,2,25/3])])
 
gp: E = ellinit([Polrev([4/3,-5/3,0,1/3]),Polrev([-3,0,1,0]),Polrev([4/3,-5/3,0,1/3]),Polrev([49/3,124/3,-2,-20/3]),Polrev([-74/3,-170/3,2,25/3])], K);
 
magma: E := EllipticCurve([K![4/3,-5/3,0,1/3],K![-3,0,1,0],K![4/3,-5/3,0,1/3],K![49/3,124/3,-2,-20/3],K![-74/3,-170/3,2,25/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^3-8/3a-2/3)\) = \((1/3a^3-8/3a-2/3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((18a^3-62a^2-35a+434)\) = \((1/3a^3-8/3a-2/3)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -3486784401 \) = \(-9^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{17984592881}{59049} a^{3} - \frac{3640839358}{6561} a^{2} - \frac{36754267676}{19683} a + \frac{202884647407}{59049} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1}{4} a^{3} - a^{2} + \frac{5}{4} a + \frac{17}{4} : a^{2} + \frac{1}{2} a - \frac{39}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 189.49113105818453569547273988304136713 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.25219264024522 \)
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\((j)_{-}\)
\((1/3a^3-8/3a-2/3)\) \(9\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(5\) 5Nn.2.2[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 9.1-a consists of curves linked by isogenies of degree 2.

Base change

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

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