Properties

Base field 5.5.36497.1
Label 5.5.36497.1-39.1-b7
Conductor \((39,2 a^{4} - 3 a^{3} - 7 a^{2} + 6 a + 2)\)
Conductor norm \( 39 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 2 \)
Rank not available

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Base field 5.5.36497.1

Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1)
gp (2.8): K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);

Weierstrass equation

\( y^2 + \left(a^{2} - a - 1\right) x y + \left(a^{2} - a - 1\right) y = x^{3} + \left(-a^{2} + a + 2\right) x^{2} + \left(10238 a^{4} - 25455 a^{3} - 18068 a^{2} + 59680 a - 20375\right) x + 897162 a^{4} - 2245513 a^{3} - 1558277 a^{2} + 5265005 a - 1771850 \)
magma: E := ChangeRing(EllipticCurve([a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850]),K);
sage: E = EllipticCurve(K, [a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850])
gp (2.8): E = ellinit([a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((39,2 a^{4} - 3 a^{3} - 7 a^{2} + 6 a + 2)\) = \( \left(a^{2} - 1\right) \cdot \left(a^{3} - 3 a\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 39 \) = \( 3 \cdot 13 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((1255737557015654093436832343547201,a + 170040169446919936099032081033994,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 724812017186553703397983614715136,a^{4} - a^{3} - 4 a^{2} + 2 a + 1020525238934952913433544527113438,a^{2} - a + 1201104003246170743075378183277650)\) = \( \left(a^{2} - 1\right)^{4} \cdot \left(a^{3} - 3 a\right)^{28} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1255737557015654093436832343547201 \) = \( 3^{4} \cdot 13^{28} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{719563077579420768826778357723754883977964219744702}{1255737557015654093436832343547201} a^{4} - \frac{381278393594661046595984821496687550631447920681206}{418579185671884697812277447849067} a^{3} - \frac{876030344833436911152144359572205314535143035049625}{418579185671884697812277447849067} a^{2} + \frac{2519311520172502928352178004640753313415969058351782}{1255737557015654093436832343547201} a + \frac{1753426510580954709523077468627597477978011837927249}{1255737557015654093436832343547201} \)
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 rank and generators

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: $\left(-\frac{241}{4} a^{4} + \frac{273}{2} a^{3} + \frac{533}{4} a^{2} - \frac{635}{2} a + \frac{351}{4} : -\frac{115}{8} a^{4} + \frac{229}{8} a^{3} + 24 a^{2} - \frac{369}{8} a + \frac{25}{4} : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} - 1\right) \) 3 \(4\) \( I_{4} \) Split multiplicative 1 4 4
\( \left(a^{3} - 3 a\right) \) 13 \(2\) \( I_{28} \) Non-split multiplicative 1 28 28

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 7, 14 and 28.
Its isogeny class 39.1-b consists of curves linked by isogenies of degrees dividing 28.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It has not yet been determined whether or not it is a \(\Q\)-curve.