Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-b1
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 5.5.65657.1

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

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

Weierstrass equation

\( y^2 + \left(-a^{4} + a^{3} + 5 a^{2} - 3 a - 3\right) x y + \left(a^{3} - 4 a - 1\right) y = x^{3} + \left(-a^{4} + 5 a^{2} - 2\right) x^{2} + \left(-527 a^{4} + 1330 a^{3} + 662 a^{2} - 1670 a - 937\right) x + 18278 a^{4} - 41440 a^{3} - 28524 a^{2} + 47105 a + 27660 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + a^3 + 5*a^2 - 3*a - 3, -a^4 + 5*a^2 - 2, a^3 - 4*a - 1, -527*a^4 + 1330*a^3 + 662*a^2 - 1670*a - 937, 18278*a^4 - 41440*a^3 - 28524*a^2 + 47105*a + 27660]),K);
sage: E = EllipticCurve(K, [-a^4 + a^3 + 5*a^2 - 3*a - 3, -a^4 + 5*a^2 - 2, a^3 - 4*a - 1, -527*a^4 + 1330*a^3 + 662*a^2 - 1670*a - 937, 18278*a^4 - 41440*a^3 - 28524*a^2 + 47105*a + 27660])
gp (2.8): E = ellinit([-a^4 + a^3 + 5*a^2 - 3*a - 3, -a^4 + 5*a^2 - 2, a^3 - 4*a - 1, -527*a^4 + 1330*a^3 + 662*a^2 - 1670*a - 937, 18278*a^4 - 41440*a^3 - 28524*a^2 + 47105*a + 27660],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((45,-a^{3} + 4 a)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 45 \) = \( 3^{2} \cdot 5 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((91125,a + 22943,a^{4} - a^{3} - 4 a^{2} + 2 a + 81824,-a^{4} + a^{3} + 5 a^{2} - 3 a + 34234,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 10336)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{6} \cdot \left(-a^{2} + a + 2\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 91125 \) = \( 3^{6} \cdot 5^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \)
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: Trivial
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]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(1\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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