Properties

Base field 6.6.434581.1
Label 6.6.434581.1-29.2-a1
Conductor \((29,-2 a^{5} + 4 a^{4} + 7 a^{3} - 8 a^{2} - 4 a + 2)\)
Conductor norm \( 29 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);

Weierstrass equation

\( y^2 + \left(a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + a\right) x y + \left(a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a - 2\right) y = x^{3} + \left(a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\right) x^{2} + \left(105 a^{5} - 194 a^{4} - 403 a^{3} + 388 a^{2} + 317 a - 39\right) x + 274 a^{5} - 510 a^{4} - 1086 a^{3} + 1108 a^{2} + 922 a - 292 \)
magma: E := ChangeRing(EllipticCurve([a^5 - 2*a^4 - 3*a^3 + 4*a^2 + a, a^4 - 2*a^3 - 2*a^2 + 3*a + 1, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 2, 105*a^5 - 194*a^4 - 403*a^3 + 388*a^2 + 317*a - 39, 274*a^5 - 510*a^4 - 1086*a^3 + 1108*a^2 + 922*a - 292]),K);
sage: E = EllipticCurve(K, [a^5 - 2*a^4 - 3*a^3 + 4*a^2 + a, a^4 - 2*a^3 - 2*a^2 + 3*a + 1, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 2, 105*a^5 - 194*a^4 - 403*a^3 + 388*a^2 + 317*a - 39, 274*a^5 - 510*a^4 - 1086*a^3 + 1108*a^2 + 922*a - 292])
gp (2.8): E = ellinit([a^5 - 2*a^4 - 3*a^3 + 4*a^2 + a, a^4 - 2*a^3 - 2*a^2 + 3*a + 1, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 2, 105*a^5 - 194*a^4 - 403*a^3 + 388*a^2 + 317*a - 39, 274*a^5 - 510*a^4 - 1086*a^3 + 1108*a^2 + 922*a - 292],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((29,-2 a^{5} + 4 a^{4} + 7 a^{3} - 8 a^{2} - 4 a + 2)\) = \( \left(3 a^{5} - 7 a^{4} - 9 a^{3} + 17 a^{2} + 3 a - 5\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 29 \) = \( 29 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((20511149,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 19107601,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 2488005,a + 18362758,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 12997129,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 14093811)\) = \( \left(3 a^{5} - 7 a^{4} - 9 a^{3} + 17 a^{2} + 3 a - 5\right)^{5} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 20511149 \) = \( 29^{5} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{23990737415377332964630580910385}{20511149} a^{5} - \frac{83448470178538726713593050182342}{20511149} a^{4} + \frac{27404100394439215638449912012936}{20511149} a^{3} + \frac{79440505494694377085636578668150}{20511149} a^{2} - \frac{21478877574700214871390186433119}{20511149} a - \frac{16227917704608257854366710709033}{20511149} \)
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\((j)_{-}\)
\( \left(3 a^{5} - 7 a^{4} - 9 a^{3} + 17 a^{2} + 3 a - 5\right) \) \(29\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 29.2-a 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.