Properties

Base field 4.4.19821.1
Label 4.4.19821.1-57.1-d1
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 yes
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 \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)
gp (2.8): K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

Weierstrass equation

\( y^2 + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) x^{2} + \left(305 a^{3} + 31 a^{2} - 2403 a - 828\right) x - 3302 a^{3} - 350 a^{2} + 26036 a + 8964 \)
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, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964]),K);
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, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964])
gp (2.8): 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, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 57 \) = \( 3 \cdot 19 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((46619598843,129140163 a + 20145865428,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 108143056 a + 29553442726,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 89208332 a + 10003103333)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{34} \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 6020462593579631409 \) = \( 3^{34} \cdot 19^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{4957694999417395}{46619598843} a^{3} + \frac{9593307021539665}{46619598843} a^{2} - \frac{11878525618289761}{46619598843} a - \frac{5175805466469244}{46619598843} \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
Generator: $\left(-\frac{5}{4} a^{3} + \frac{1}{4} a^{2} + \frac{37}{4} a + \frac{5}{2} : -\frac{5}{12} a^{3} - \frac{5}{12} a^{2} + \frac{31}{8} a + \frac{3}{8} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \) \(3\) \(2\) \( I_{34} \) Non-split multiplicative \(1\) \(34\) \(34\)
\( \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right) \) \(19\) \(2\) \( I_{2} \) Split multiplicative \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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 degree2.

Base change

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