Base field \(\Q(\sqrt{69}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 17 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-17, -1, 1]))
gp: K = nfinit(Polrev([-17, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,1]),K([-242,53]),K([1064,-231])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,1]),Polrev([-242,53]),Polrev([1064,-231])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,1],K![-242,53],K![1064,-231]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a-1)\) | = | \((a+4)\cdot(-a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((253513a+1124803)\) | = | \((a+4)\cdot(-a+4)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 457763671875 \) | = | \(3\cdot5^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{319450720644289}{457763671875} a + \frac{2560911940710052}{457763671875} \) | ||
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: | $\left(3 a - \frac{53}{4} : -2 a + \frac{49}{8} : 1\right)$ |
| 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: | \( 5.1314535786739268131219904983227953439 \) | ||
| Tamagawa product: | \( 16 \) = \(1\cdot2^{4}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.4710176665547358417915822526818902812 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+4)\) | \(3\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((-a+4)\) | \(5\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
15.1-c
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.