Base field \(\Q(\sqrt{-77}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 77 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((37,a+16)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{7}{3} a - \frac{167}{9} : \frac{250}{27} a + \frac{835}{9} : 1\right)$ | $1.4944407538016703049134574705638987676$ | $\infty$ |
| $\left(\frac{5}{2} a - 20 : \frac{33}{4} a + \frac{423}{4} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((8)\) | = | \((2,a+1)^{6}\) |
|
| |||||
| Conductor norm: | $N(\frak{N})$ | = | \( 64 \) | = | \(2^{6}\) |
|
| |||||
| Discriminant: | $\Delta$ | = | $2337510240a+764500472$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((2337510240a+764500472)\) | = | \((2,a+1)^{6}\cdot(37,a+16)^{12}\) |
|
| |||||
| Discriminant norm: | $N(\Delta)$ | = | \( 421308928373762257984 \) | = | \(2^{6}\cdot37^{12}\) |
|
| |||||
| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((8)\) | = | \((2,a+1)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 64 \) | = | \(2^{6}\) |
| j-invariant: | $j$ | = | \( 287496 \) | ||
|
| |||||
| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-4}]\) (potential complex multiplication) | ||
|
| |||||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
|
|
|||
| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.4944407538016703049134574705638987676 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.9888815076033406098269149411277975352 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 13.750371636040745654980191559621114396 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.3417890769269479132720779108806583694 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}2.341789077 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 13.750372 \cdot 2.988882 \cdot 1 } { {2^2 \cdot 17.549929} } \\ & \approx 2.341789077 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(1\) | \(II\) | Additive | \(1\) | \(6\) | \(6\) | \(0\) |
| \((37,a+16)\) | \(37\) | \(1\) | \(I_0\) | 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 |
|---|---|
| \(2\) | 2Cs |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
64.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 1568.e1 |
| \(\Q\) | 3872.f2 |