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 \((23,a+3)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{5}{2} a + 57 : -\frac{109}{4} a - \frac{759}{4} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((12,2a+6)\) | = | \((2,a+1)^{3}\cdot(3,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 24 \) | = | \(2^{3}\cdot3\) |
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| Discriminant: | $\Delta$ | = | $116926467840a+31052045456928$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((116926467840a+31052045456928)\) | = | \((2,a+1)^{10}\cdot(3,a)^{16}\cdot(23,a+3)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 965993189114841317803459584 \) | = | \(2^{10}\cdot3^{16}\cdot23^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((209952)\) | = | \((2,a+1)^{10}\cdot(3,a)^{16}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 44079842304 \) | = | \(2^{10}\cdot3^{16}\) |
| j-invariant: | $j$ | = | \( \frac{207646}{6561} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 0 \) |
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| Mordell-Weil rank: | $r$ | = | \(0\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | = | \( 1 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | = | \( 1 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 7.2706940358628777611249866685015359950 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 0.64014891534917783508745696881920096456 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}0.640148915 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 3.635347 \cdot 1 \cdot 4 } { {2^2 \cdot 22.715633} } \\ & \approx 0.640148915 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((3,a)\) | \(3\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
| \((23,a+3)\) | \(23\) | \(1\) | \(I_0\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
24.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 72.a6 |
| \(\Q\) | 88752.p6 |