Base field \(\Q(\sqrt{-129}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 129 \); class number \(12\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(12 : -6 a + 2 : 1\right)$ | $0.70155617528104437603827294535073893312$ | $\infty$ |
| $\left(-\frac{923449}{972} : -\frac{112697107}{52488} a : 1\right)$ | $13.748613869349245719372881381028539722$ | $\infty$ |
| $\left(8 : -4 a : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((14,7a+7)\) | = | \((2,a+1)\cdot(7,a+2)\cdot(7,a+5)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 98 \) | = | \(2\cdot7\cdot7\) |
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| Discriminant: | $\Delta$ | = | $20412$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((20412)\) | = | \((2,a+1)^{4}\cdot(3,a)^{12}\cdot(7,a+2)\cdot(7,a+5)\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 416649744 \) | = | \(2^{4}\cdot3^{12}\cdot7\cdot7\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((28)\) | = | \((2,a+1)^{4}\cdot(7,a+2)\cdot(7,a+5)\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 784 \) | = | \(2^{4}\cdot7\cdot7\) |
| j-invariant: | $j$ | = | \( -\frac{15625}{28} \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 9.6454249615965771723207662969449374797 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 38.581699846386308689283065187779749919 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 31.515016866984194237974259626482832830 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2^{2}\cdot1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 26.763570729112563690095723452249100552 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}26.763570729 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 15.757508 \cdot 38.581700 \cdot 4 } { {2^2 \cdot 22.715633} } \\ & \approx 26.763570729 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((3,a)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((7,a+2)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((7,a+5)\) | \(7\) | \(1\) | \(I_{1}\) | 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 |
| \(3\) | 3Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
98.2-d
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 1008.h5 |
| \(\Q\) | 25886.d5 |