Base field \(\Q(\sqrt{-399}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 100 \); class number \(16\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a+1)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(4 : -7 : 1\right)$ | $0.43836053379150852116146504588044365523$ | $\infty$ |
| $\left(-\frac{9567}{61504} a - \frac{65213}{15376} : \frac{6521505}{15252992} a - \frac{28848989}{3813248} : 1\right)$ | $5.5144924207955317444632675438207029136$ | $\infty$ |
| $\left(-5 : 2 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((7)\) | = | \((7,a+3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 49 \) | = | \(7^{2}\) |
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| Discriminant: | $\Delta$ | = | $250047$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((250047)\) | = | \((3,a+1)^{12}\cdot(7,a+3)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 62523502209 \) | = | \(3^{12}\cdot7^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((343)\) | = | \((7,a+3)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 117649 \) | = | \(7^{6}\) |
| j-invariant: | $j$ | = | \( -3375 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.9849759366007106813550064081418099825 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 7.9399037464028427254200256325672399300 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 9.8890092005650934729963939363687552888 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(1\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.9308056418278266154298593571172150661 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.930805642 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 9.889009 \cdot 7.939904 \cdot 4 } { {2^2 \cdot 19.974984} } \\ & \approx 3.930805642 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a+1)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((7,a+3)\) | \(7\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.6.2 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 441.c4 |
| \(\Q\) | 17689.e2 |