Base field \(\Q(\sqrt{-102}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 102 \); class number \(4\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((17,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{490}{3} : -\frac{1904}{9} a + \frac{245}{3} : 1\right)$ | $1.1613532645249444617186879256722155890$ | $\infty$ |
| $\left(4626 : 312375 : 1\right)$ | $7.4039624769253249188223145184033556303$ | $\infty$ |
| $\left(18 : -9 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((14,7a)\) | = | \((2,a)\cdot(7)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 98 \) | = | \(2\cdot49\) |
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| Discriminant: | $\Delta$ | = | $529867914688$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((529867914688)\) | = | \((2,a)^{12}\cdot(17,a)^{12}\cdot(7)^{3}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 280760007015809646137344 \) | = | \(2^{12}\cdot17^{12}\cdot49^{3}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((21952)\) | = | \((2,a)^{12}\cdot(7)^{3}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 481890304 \) | = | \(2^{12}\cdot49^{3}\) |
| j-invariant: | $j$ | = | \( \frac{9938375}{21952} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 8.5986159929974198757697580630397549227 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 34.394463971989679503079032252159019691 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 5.2525028111640323729957099377471388050 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 6 \) = \(2\cdot1\cdot3\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 13.415782739147892641008178028847785637 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}13.415782739 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 5.252503 \cdot 34.394464 \cdot 6 } { {2^2 \cdot 20.199010} } \\ & \approx 13.415782739 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((17,a)\) | \(17\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((7)\) | \(49\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
| \(3\) | 3Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
98.1-b
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 4032.w6 |
| \(\Q\) | 4046.f6 |