Base field \(\Q(\sqrt{-95}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 24 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((3,a+2)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{4183}{5808} a + \frac{12077}{2178} : -\frac{6600491}{2299968} a - \frac{3642967}{287496} : 1\right)$ | $8.1429708281135836641552716279096076023$ | $\infty$ |
| $\left(-2 : -a - 2 : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((19,a+9)\) | = | \((19,a+9)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 19 \) | = | \(19\) |
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| Discriminant: | $\Delta$ | = | $-446a-2095$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-446a-2095)\) | = | \((3,a+2)^{12}\cdot(19,a+9)\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 10097379 \) | = | \(3^{12}\cdot19\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((19,a+9)\) | = | \((19,a+9)\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 19 \) | = | \(19\) |
| j-invariant: | $j$ | = | \( \frac{86022}{19} a + \frac{1125603}{19} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 8.1429708281135836641552716279096076023 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 16.28594165622716732831054325581921520 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 17.19003450511886627829664653147925633 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.7951793245745404644651257067617937314 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.795179325 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 17.190035 \cdot 16.285942 \cdot 1 } { {4^2 \cdot 9.746794} } \\ & \approx 1.795179325 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a+2)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((19,a+9)\) | \(19\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
19.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.