Properties

Base field 4.4.19821.1
Label 4.4.19821.1-57.1-d2
Conductor \((57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)\)
Conductor norm \( 57 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 4.4.19821.1

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

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)
 
gp: K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) x^{2} + \left(-\frac{190}{3} a^{3} - \frac{22}{3} a^{2} + 502 a + 172\right) x - \frac{1208}{3} a^{3} - \frac{134}{3} a^{2} + 3179 a + 1095 \)
sage: E = EllipticCurve(K, [-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -190/3*a^3 - 22/3*a^2 + 502*a + 172, -1208/3*a^3 - 134/3*a^2 + 3179*a + 1095])
 
gp: E = ellinit([-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -190/3*a^3 - 22/3*a^2 + 502*a + 172, -1208/3*a^3 - 134/3*a^2 + 3179*a + 1095],K)
 
magma: E := ChangeRing(EllipticCurve([-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -190/3*a^3 - 22/3*a^2 + 502*a + 172, -1208/3*a^3 - 134/3*a^2 + 3179*a + 1095]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 57 \) = \( 3 \cdot 19 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((373977,6561 a + 275562,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 4654 a + 48082,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 4976 a + 301148)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{17} \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 2453663097 \) = \( 3^{17} \cdot 19 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{217217028070}{373977} a^{3} + \frac{1324842416596}{373977} a^{2} - \frac{2515509051217}{373977} a + \frac{161738260330}{41553} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1 : -a + 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \) \(3\) \(1\) \(I_{17}\) Non-split multiplicative \(1\) \(1\) \(17\) \(17\)
\( \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right) \) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 57.1-d consists of curves linked by isogenies of degree 2.

Base change

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