Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Invariants
| Conductor: | \((-77a+32)\) | = | \((9a-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 4489 \) | = | \(67^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((58853475a+1319735968)\) | = | \((9a-2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1822837804551761449 \) | = | \(67^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 0 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (complex multiplication) | |
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{U}(1)$ | ||
Mordell-Weil group
| Rank: | \(0 \le r \le 2\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0 \le r \le 2\) | ||
| Regulator*: | \( 1 \) | ||
| Period: | \( 0.42245360059058326808010931151921160509 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.9512296001687991653966573082374851614 \) | ||
| Analytic order of Ш*: | \( 4 \) (rounded) | ||
* Conditional on BSD: assuming rank = analytic rank.
Note: We expect that the nontriviality of Ш explains the discrepancy between the upper bound on the rank and the analytic rank. The application of further descents should suffice to establish the weak BSD conjecture for this curve.
Local data at primes of bad reduction
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((9a-2)\) | \(67\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(2\) | \(10\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(67\) | 67Cs.9.1 |
For all other primes \(p\), the image is a Borel subgroup if \(p=3\), a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 4489.3-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.