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 \((2,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-a + 6 : -3 a - 14 : 1\right)$ | $0.49796948996928521624651937326166960829$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((18,2a+6)\) | = | \((2,a)\cdot(2,a+1)\cdot(3,a)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 36 \) | = | \(2\cdot2\cdot3^{2}\) |
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| Discriminant: | $\Delta$ | = | $-2912a-22272$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-2912a-22272)\) | = | \((2,a)^{15}\cdot(2,a+1)^{5}\cdot(3,a)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 764411904 \) | = | \(2^{15}\cdot2^{5}\cdot3^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((23328,8a+19608)\) | = | \((2,a)^{3}\cdot(2,a+1)^{5}\cdot(3,a)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 186624 \) | = | \(2^{3}\cdot2^{5}\cdot3^{6}\) |
| j-invariant: | $j$ | = | \( -\frac{8045895}{32} a + \frac{10335215}{8} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.49796948996928521624651937326166960829 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.9959389799385704324930387465233392166 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 6.004966243860786838796415305890031771 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 5 \) = \(1\cdot5\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.0679727836165191673979513707104123046 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.067972784 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 6.004966 \cdot 0.995939 \cdot 5 } { {1^2 \cdot 9.746794} } \\ & \approx 3.067972784 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((2,a+1)\) | \(2\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((3,a)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
36.4-a
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.