Base field \(\Q(\sqrt{-165}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 165 \); 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
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{27}{2} : -\frac{27}{4} a + \frac{45}{4} : 1\right)$ | $0.65015642199656306277411657149174565903$ | $\infty$ |
| $\left(\frac{259}{15} : -\frac{1936}{225} a : 1\right)$ | $3.7751275877018922740978548852251980532$ | $\infty$ |
| $\left(6 : -3 a : 1\right)$ | $0$ | $2$ |
| $\left(\frac{69}{4} : -\frac{69}{8} a : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((15,a)\) | = | \((3,a)\cdot(5,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 15 \) | = | \(3\cdot5\) |
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| Discriminant: | $\Delta$ | = | $164025$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((164025)\) | = | \((3,a)^{16}\cdot(5,a)^{4}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 26904200625 \) | = | \(3^{16}\cdot5^{4}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((225)\) | = | \((3,a)^{4}\cdot(5,a)^{4}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 50625 \) | = | \(3^{4}\cdot5^{4}\) |
| j-invariant: | $j$ | = | \( \frac{13997521}{225} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.4544234450007786070182724689601753184 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 9.817693780003114428073089875840701274 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 8.942806850470116407066971128448966889 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2^{2}\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 6.8350447260452489607818001656479041800 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 4 \) (rounded) |
BSD formula
$$\begin{aligned}6.835044726 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 8.942807 \cdot 9.817694 \cdot 8 } { {4^2 \cdot 25.690465} } \\ & \approx 6.835044726 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((5,a)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
15.1-c
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 720.c6 |
| \(\Q\) | 9075.g6 |