Properties

Label 4.4.5125.1-55.2-c3
Base field 4.4.5125.1
Conductor \((-3a^2+4a+10)\)
Conductor norm \( 55 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.5125.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-4a-5\right){x}^{2}+\left(-155a^{3}+482a^{2}+395a-1557\right){x}-2048a^{3}+6363a^{2}+5087a-19778\)
sage: E = EllipticCurve([K([1,-4,-1,1]),K([-5,-4,0,1]),K([0,1,0,0]),K([-1557,395,482,-155]),K([-19778,5087,6363,-2048])])
 
gp: E = ellinit([Pol(Vecrev([1,-4,-1,1])),Pol(Vecrev([-5,-4,0,1])),Pol(Vecrev([0,1,0,0])),Pol(Vecrev([-1557,395,482,-155])),Pol(Vecrev([-19778,5087,6363,-2048]))], K);
 
magma: E := EllipticCurve([K![1,-4,-1,1],K![-5,-4,0,1],K![0,1,0,0],K![-1557,395,482,-155],K![-19778,5087,6363,-2048]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a^2+4a+10)\) = \((-a^3+a^2+4a+1)\cdot(a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 55 \) = \(5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^3+12a^2+14a-34)\) = \((-a^3+a^2+4a+1)^{3}\cdot(a-1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 166375 \) = \(5^{3}\cdot11^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{133990069789633150033955762628}{6655} a^{3} + \frac{496615724716951900546923293583}{6655} a^{2} - \frac{8693172568676612045861200086}{1331} a - \frac{78523816804016589414106934741}{605} \)
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: $\left(-\frac{13}{4} a^{3} + \frac{35}{4} a^{2} + \frac{33}{4} a - 22 : \frac{19}{8} a^{3} - \frac{19}{2} a^{2} - \frac{35}{8} a + 33 : 1\right)$
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: \( 117.381814862661 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.63966031344996 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a-1)\) \(11\) \(1\) \(I_{3}\) 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
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 55.2-c consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.