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 \((3,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{188}{121} a - \frac{53}{121} : \frac{1282}{1331} a - \frac{81181}{1331} : 1\right)$ | $1.5937243877619066838001532676555451253$ | $\infty$ |
| $\left(-\frac{8}{9} a + \frac{11}{3} : -\frac{37}{27} a + \frac{46}{9} : 1\right)$ | $1.6527739206305112096362821571841366984$ | $\infty$ |
| $\left(-\frac{7}{4} a - \frac{1}{2} : 2 a - \frac{523}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((10)\) | = | \((2)\cdot(5,a)\cdot(5,a+4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 100 \) | = | \(4\cdot5\cdot5\) |
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| Discriminant: | $\Delta$ | = | $-7000000a+71625000$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-7000000a+71625000)\) | = | \((3,a)^{12}\cdot(2)^{3}\cdot(5,a)^{6}\cdot(5,a+4)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 8303765625000000 \) | = | \(3^{12}\cdot4^{3}\cdot5^{6}\cdot5^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((125000)\) | = | \((2)^{3}\cdot(5,a)^{6}\cdot(5,a+4)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 15625000000 \) | = | \(4^{3}\cdot5^{6}\cdot5^{6}\) |
| j-invariant: | $j$ | = | \( \frac{10260751717}{125000} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.4093500821856572596085400783044712758 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 9.6374003287426290384341603132178851032 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.0524160717343869918904245502580096090 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 12 \) = \(1\cdot1\cdot( 2 \cdot 3 )\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 5.1037488099607731714153652571896757100 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}5.103748810 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.052416 \cdot 9.637400 \cdot 12 } { {2^2 \cdot 17.291616} } \\ & \approx 5.103748810 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((2)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((5,a)\) | \(5\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((5,a+4)\) | \(5\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
| \(3\) | 3Ns |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
100.2-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.