Base field \(\Q(\sqrt{-26}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 26 \); class number \(6\).
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
trivial
Invariants
| Conductor: | $\frak{N}$ | = | \((28,14a)\) | = | \((2,a)^{3}\cdot(7,a+3)\cdot(7,a+4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 392 \) | = | \(2^{3}\cdot7\cdot7\) |
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| Discriminant: | $\Delta$ | = | $-11200a+58352$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-11200a+58352)\) | = | \((2,a)^{8}\cdot(3,a+1)^{12}\cdot(7,a+3)\cdot(7,a+4)\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 6666395904 \) | = | \(2^{8}\cdot3^{12}\cdot7\cdot7\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((112)\) | = | \((2,a)^{8}\cdot(7,a+3)\cdot(7,a+4)\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 12544 \) | = | \(2^{8}\cdot7\cdot7\) |
| j-invariant: | $j$ | = | \( \frac{35680}{7} a + \frac{169412}{7} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 10.469982312230551762029957948754856265 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) = \(2\cdot1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.0533324660398034105654380207914842069 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.053332466 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 10.469982 \cdot 1 \cdot 2 } { {1^2 \cdot 10.198039} } \\ & \approx 2.053332466 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(3\) | \(8\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((7,a+3)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((7,a+4)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 392.2-d consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.