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
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{23}{8} : -\frac{27}{32} a + \frac{15}{16} : 1\right)$ | $1.4713342432646010310106195184556460465$ | $\infty$ |
| $\left(-4 : 6 : 1\right)$ | $0$ | $6$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((30,15a)\) | = | \((2,a)\cdot(3,a+1)\cdot(3,a+2)\cdot(5,a+2)\cdot(5,a+3)\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 450 \) | = | \(2\cdot3\cdot3\cdot5\cdot5\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $5314410$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((5314410)\) | = | \((2,a)^{2}\cdot(3,a+1)^{12}\cdot(3,a+2)^{12}\cdot(5,a+2)\cdot(5,a+3)\) |
|
| |||||
| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 28242953648100 \) | = | \(2^{2}\cdot3^{12}\cdot3^{12}\cdot5\cdot5\) |
|
| |||||
| j-invariant: | $j$ | = | \( \frac{35578826569}{5314410} \) | ||
|
| |||||
| 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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.4713342432646010310106195184556460465 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.9426684865292020620212390369112920930 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.94143521051915969549575008322322966336 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 288 \) = \(2\cdot( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(6\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.4816460869807661772285437750933795648 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.481646087 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.941435 \cdot 2.942668 \cdot 288 } { {6^2 \cdot 10.198039} } \\ & \approx 4.481646087 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 5 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((3,a+1)\) | \(3\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
| \((3,a+2)\) | \(3\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
| \((5,a+2)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((5,a+3)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
450.5-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 30.a5 |
| \(\Q\) | 162240.ei5 |