Base field \(\Q(\sqrt{-210}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 210 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((29,a+14)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{7}{4} a + \frac{133}{8} : -\frac{293}{32} a + \frac{3157}{16} : 1\right)$ | $1.4944407538016703049134574705638987676$ | $\infty$ |
| $\left(-\frac{11}{2} a + \frac{81}{4} : 17 a - \frac{1785}{8} : 1\right)$ | $2.6964694026626170544163641927814947238$ | $\infty$ |
| $\left(2 a + \frac{33}{2} : -\frac{33}{4} a + 210 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((8,4a)\) | = | \((2,a)^{5}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 32 \) | = | \(2^{5}\) |
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| Discriminant: | $\Delta$ | = | $-67468896a-4657060792$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-67468896a-4657060792)\) | = | \((2,a)^{6}\cdot(29,a+14)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 22644146125150018624 \) | = | \(2^{6}\cdot29^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((8)\) | = | \((2,a)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 64 \) | = | \(2^{6}\) |
| j-invariant: | $j$ | = | \( 287496 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-4}]\) (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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 4.0297137667182610839749820232413065756 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 16.118855066873044335899928092965226302 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 27.500743272081491309960383119242228792 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 15.294630137558454753995654592581041517 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}15.294630138 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 13.750372 \cdot 16.118855 \cdot 2 } { {2^2 \cdot 28.982753} } \\ & \approx 15.294630138 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(5\) | \(6\) | \(0\) |
| \((29,a+14)\) | \(29\) | \(1\) | \(I_0\) | 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
32.1-e
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.e2 |
| \(\Q\) | 14400.cx1 |