Base field \(\Q(\sqrt{-127}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 32 \); class number \(5\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((11,a+3)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-4 a + 4 : -4 a + 8 : 1\right)$ | $0.50170751437424876501702107429828534194$ | $\infty$ |
| $\left(-\frac{1}{4} a - 4 : -\frac{15}{8} a + 68 : 1\right)$ | $1.0345812015299418947759587929110789671$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((16,2a)\) | = | \((2,a)^{4}\cdot(2,a+1)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 32 \) | = | \(2^{4}\cdot2\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $6952329216a-46315741184$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((6952329216a-46315741184)\) | = | \((2,a)^{13}\cdot(2,a+1)^{17}\cdot(11,a+3)^{12}\) |
|
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 3369861809713765679104 \) | = | \(2^{13}\cdot2^{17}\cdot11^{12}\) |
|
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((131072,8192a+122880)\) | = | \((2,a)^{13}\cdot(2,a+1)^{17}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 1073741824 \) | = | \(2^{13}\cdot2^{17}\) |
| j-invariant: | $j$ | = | \( -\frac{6357609}{131072} a + \frac{1888137}{4096} \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.51459582378174249951123877566468109714 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.05838329512696999804495510265872438856 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 4.9038599127421478235121164585058372152 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2^{2}\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.5827989218207640432359202270574525651 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.582798922 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 4.903860 \cdot 2.058383 \cdot 4 } { {1^2 \cdot 11.269428} } \\ & \approx 3.582798922 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(I_{17}\) | Non-split multiplicative | \(1\) | \(1\) | \(17\) | \(17\) |
| \((11,a+3)\) | \(11\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(17\) | 17B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
17.
Its isogeny class
32.2-a
consists of curves linked by isogenies of
degree 17.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.