Base field \(\Q(\sqrt{-95}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 24 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((11,a+6)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{839}{121} a - \frac{3621}{605} : \frac{1548594}{33275} a - \frac{545472}{33275} : 1\right)$ | $5.0552731943798577233444438524638954488$ | $\infty$ |
| $\left(-3 a - 9 : 0 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((24)\) | = | \((2,a)^{3}\cdot(2,a+1)^{3}\cdot(3,a)\cdot(3,a+2)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 576 \) | = | \(2^{3}\cdot2^{3}\cdot3\cdot3\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $-386876030976a+2048404432896$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-386876030976a+2048404432896)\) | = | \((2,a)^{11}\cdot(2,a+1)^{11}\cdot(3,a)^{6}\cdot(3,a+2)^{6}\cdot(11,a+6)^{12}\) |
|
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 6995635664125364633862144 \) | = | \(2^{11}\cdot2^{11}\cdot3^{6}\cdot3^{6}\cdot11^{12}\) |
|
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((1492992)\) | = | \((2,a)^{11}\cdot(2,a+1)^{11}\cdot(3,a)^{6}\cdot(3,a+2)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 2229025112064 \) | = | \(2^{11}\cdot2^{11}\cdot3^{6}\cdot3^{6}\) |
| j-invariant: | $j$ | = | \( \frac{2060602}{729} \) | ||
|
| |||||
| 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: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 5.0552731943798577233444438524638954488 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 10.11054638875971544668888770492779090 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.512968670319328336599936295084837081 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 12 \) = \(1\cdot1\cdot( 2 \cdot 3 )\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 7.8202592819541140004411613009860986510 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}7.820259282 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.512969 \cdot 10.110546 \cdot 12 } { {2^2 \cdot 9.746794} } \\ & \approx 7.820259282 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(3\) | \(11\) | \(0\) |
| \((3,a)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((3,a+2)\) | \(3\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((11,a+6)\) | \(11\) | \(1\) | \(I_0\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
576.11-c
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.