Base field \(\Q(\sqrt{-299}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 75 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((7,a+1)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{5}{4} a - \frac{165}{4} : \frac{85}{4} a - \frac{375}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((9,3a)\) | = | \((3,a)^{2}\cdot(3,a+2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 27 \) | = | \(3^{2}\cdot3\) |
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| Discriminant: | $\Delta$ | = | $2580840a+16582023$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((2580840a+16582023)\) | = | \((3,a)^{8}\cdot(3,a+2)^{2}\cdot(7,a+1)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 817314167931849 \) | = | \(3^{8}\cdot3^{2}\cdot7^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((6561,9a+1431)\) | = | \((3,a)^{8}\cdot(3,a+2)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 59049 \) | = | \(3^{8}\cdot3^{2}\) |
| j-invariant: | $j$ | = | \( \frac{12214672127}{9} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 3.1791739331530618770531201104435698472 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.6188295246969263506152562671605413718 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 36 \) (rounded) |
BSD formula
$$\begin{aligned}6.618829525 \approx L(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 36 \cdot 3.179174 \cdot 1 \cdot 4 } { {2^2 \cdot 17.291616} } \\ & \approx 6.618829525 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(2\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((3,a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((7,a+1)\) | \(7\) | \(1\) | \(I_0\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
27.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.