Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-1,1,0,0,0]),K([-1,-3,2,2,-1,0]),K([-1,-1,1,0,0,0]),K([-14,48,-35,-26,13,0]),K([-32,145,-106,-78,39,0])])
gp: E = ellinit([Polrev([-1,-1,1,0,0,0]),Polrev([-1,-3,2,2,-1,0]),Polrev([-1,-1,1,0,0,0]),Polrev([-14,48,-35,-26,13,0]),Polrev([-32,145,-106,-78,39,0])], K);
magma: E := EllipticCurve([K![-1,-1,1,0,0,0],K![-1,-3,2,2,-1,0],K![-1,-1,1,0,0,0],K![-14,48,-35,-26,13,0],K![-32,145,-106,-78,39,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+5a^4+6a^3-14a^2-3a+4)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((40a^4-80a^3-175a^2+215a-156)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{16}\cdot(a^2-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 248155780267521 \) | = | \(3^{16}\cdot7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2469125117705}{15752961} a^{4} - \frac{4938250235410}{15752961} a^{3} - \frac{10256295035167}{15752961} a^{2} + \frac{605972388232}{750141} a + \frac{12380135655164}{15752961} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) | |
| Generator | $\left(-7 a^{5} + 9 a^{4} + 30 a^{3} - 12 a^{2} - 22 a + 9 : -50 a^{5} + 60 a^{4} + 214 a^{3} - 71 a^{2} - 117 a + 34 : 1\right)$ | |
| Height | \(0.52307482150179035179576209282681854202\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{4} - 4 a^{3} - 6 a^{2} + 8 a + 1 : -a^{2} + a + 2 : 1\right)$ | $\left(\frac{3}{4} a^{4} - \frac{3}{2} a^{3} - \frac{7}{4} a^{2} + \frac{5}{2} a - \frac{5}{4} : -\frac{1}{4} a^{4} + \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{1}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.52307482150179035179576209282681854202 \) | ||
| Period: | \( 1118.9642602620879668378175197910417076 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.98186 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
| \((a^2-2)\) | \(7\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| 3.3.257.1 | 3.3.257.1-21.1-a3 |
| 3.3.257.1 | a curve with conductor norm 441 (not in the database) |