Base field \(\Q(\zeta_{13})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,-3,0,1,0]),K([-1,1,3,0,-1,0]),K([0,1,-3,0,1,0]),K([-231,85,365,73,-200,4]),K([-1732,-931,4755,1997,-2877,24])])
gp: E = ellinit([Polrev([0,1,-3,0,1,0]),Polrev([-1,1,3,0,-1,0]),Polrev([0,1,-3,0,1,0]),Polrev([-231,85,365,73,-200,4]),Polrev([-1732,-931,4755,1997,-2877,24])], K);
magma: E := EllipticCurve([K![0,1,-3,0,1,0],K![-1,1,3,0,-1,0],K![0,1,-3,0,1,0],K![-231,85,365,73,-200,4],K![-1732,-931,4755,1997,-2877,24]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-2a+3)\) | = | \((a^3-a^2-2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 79 \) | = | \(79\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((666a^5-938a^4-3298a^3+3552a^2+3555a-661)\) | = | \((a^3-a^2-2a+3)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -9468276082626847201 \) | = | \(-79^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{127794683808852859853832701635933437922415}{9468276082626847201} a^{5} + \frac{120367721300657709905855839586570070913343}{9468276082626847201} a^{4} - \frac{405233310886078181372607580308878671807916}{9468276082626847201} a^{3} - \frac{275737199471198880155191606074376860275461}{9468276082626847201} a^{2} + \frac{231318547680046130469307232906240352482422}{9468276082626847201} a + \frac{65809650751079058294984275680386783414465}{9468276082626847201} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{13}{4} a^{5} - \frac{43}{4} a^{4} - \frac{39}{4} a^{3} + \frac{127}{4} a^{2} + \frac{11}{4} a - 9 : 8 a^{5} - \frac{59}{8} a^{4} - \frac{179}{8} a^{3} + \frac{49}{2} a^{2} + \frac{3}{8} a - 3 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 2.9187900687961089692499549562740333975 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.49691 \) | ||
| Analytic order of Ш: | \( 625 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-2a+3)\) | \(79\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
79.1-c
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.