Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 5*x^3 - x^2 + 3*x + 1)
gp (2.8): K = nfinit(a^5 - 5*a^3 - a^2 + 3*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^4 - 4*a^2, -3*a^4 + 14*a^2 + a - 4, 2*a^4 - 9*a^2 + 2, 262*a^4 - 221*a^3 - 1174*a^2 + 730*a + 366, 3311*a^4 - 2134*a^3 - 14758*a^2 + 6316*a + 3956]),K);
sage: E = EllipticCurve(K, [a^4 - 4*a^2, -3*a^4 + 14*a^2 + a - 4, 2*a^4 - 9*a^2 + 2, 262*a^4 - 221*a^3 - 1174*a^2 + 730*a + 366, 3311*a^4 - 2134*a^3 - 14758*a^2 + 6316*a + 3956])
gp (2.8): E = ellinit([a^4 - 4*a^2, -3*a^4 + 14*a^2 + a - 4, 2*a^4 - 9*a^2 + 2, 262*a^4 - 221*a^3 - 1174*a^2 + 730*a + 366, 3311*a^4 - 2134*a^3 - 14758*a^2 + 6316*a + 3956],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((29,2 a^{4} - a^{3} - 10 a^{2} + 2 a + 4)\) | = | \( \left(-2 a^{4} + a^{3} + 10 a^{2} - 2 a - 4\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 29 \) | = | \( 29 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((17249876309,a^{4} - 5 a^{2} + 15330541788,-a^{4} + 5 a^{2} + a + 4936804027,2 a^{4} - a^{3} - 9 a^{2} + 3 a + 3745110535,-a^{4} + a^{3} + 5 a^{2} - 3 a + 11514498930)\) | = | \( \left(-2 a^{4} + a^{3} + 10 a^{2} - 2 a - 4\right)^{7} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 17249876309 \) | = | \( 29^{7} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{1081740781776866294940863471759}{17249876309} a^{4} + \frac{2120240724895871393224432246770}{17249876309} a^{3} + \frac{43206056991447348862819034032}{594823321} a^{2} - \frac{1374124848200655188432751002870}{17249876309} a - \frac{551901067153896026723504870757}{17249876309} \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-2 a^{4} + a^{3} + 10 a^{2} - 2 a - 4\right) \) | \(29\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
29.1-b
consists of curves linked by isogenies of
degree 7.