Base field \(\Q(\sqrt{-26}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 26 \); class number \(6\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 1 $)
Mordell-Weil generators
No non-torsion generators are known.
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(0 : -1 : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((27)\) | = | \((3,a+1)^{3}\cdot(3,a+2)^{3}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 729 \) | = | \(3^{3}\cdot3^{3}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $27$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((27)\) | = | \((3,a+1)^{3}\cdot(3,a+2)^{3}\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 729 \) | = | \(3^{3}\cdot3^{3}\) |
|
| |||||
| j-invariant: | $j$ | = | \( 0 \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r?$ | \(0 \le r \le 1\) | |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | not available |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | not available |
| Global period: | $\Omega(E/K)$ | ≈ | \( 16.217256528125389281692606040775220472 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.6591850085990534928613311559373505503 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | not available |
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a+1)\) | \(3\) | \(1\) | \(II\) | Additive | \(-1\) | \(3\) | \(3\) | \(0\) |
| \((3,a+2)\) | \(3\) | \(1\) | \(II\) | Additive | \(-1\) | \(3\) | \(3\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs.1.1 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
729.4-b
consists of curves linked by isogenies of
degrees dividing 27.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 27.a4 |
| \(\Q\) | 292032.ee3 |