Base field \(\Q(\sqrt{-95}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 24 \); class number \(8\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-7 : -10 : 1\right)$ | $0.13482717795889418687569402828090256769$ | $\infty$ |
| $\left(-19 : 15 a + 2 : 1\right)$ | $0.58900295851831389723493034516213624668$ | $\infty$ |
| $\left(-\frac{1}{4} : \frac{1}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-6a+3)\) | = | \((3,a)\cdot(3,a+2)\cdot(5,a+2)\cdot(19,a+9)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 855 \) | = | \(3\cdot3\cdot5\cdot19\) |
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| Discriminant: | $\Delta$ | = | $28048275$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((28048275)\) | = | \((3,a)^{10}\cdot(3,a+2)^{10}\cdot(5,a+2)^{4}\cdot(19,a+9)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 786705730475625 \) | = | \(3^{10}\cdot3^{10}\cdot5^{4}\cdot19^{2}\) |
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| j-invariant: | $j$ | = | \( \frac{48587168449}{28048275} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.079413606706463878539850508225694273381 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.3176544268258555141594020329027770935240 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.57753248048032160534912261785698335412 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 400 \) = \(( 2 \cdot 5 )\cdot( 2 \cdot 5 )\cdot2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.1412819246877047526352947992323611453 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.141281925 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.577532 \cdot 0.317654 \cdot 400 } { {2^2 \cdot 9.746794} } \\ & \approx 5.141281925 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
| \((3,a+2)\) | \(3\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
| \((5,a+2)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((19,a+9)\) | \(19\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
855.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 285.a1 |
| \(\Q\) | 27075.k1 |