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
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 : 2 : 1\right)$ | $1.3186317800226628430785586238185946618$ | $\infty$ |
| $\left(-\frac{588}{3481} a - \frac{782}{3481} : -\frac{19152}{205379} a + \frac{240577}{205379} : 1\right)$ | $3.9370673074528633267438286764002631788$ | $\infty$ |
| $\left(1 : -1 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((55,a)\) | = | \((5,a)\cdot(11,a)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 55 \) | = | \(5\cdot11\) |
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| Discriminant: | $\Delta$ | = | $40095$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((40095)\) | = | \((3,a)^{12}\cdot(5,a)^{2}\cdot(11,a)^{2}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 1607609025 \) | = | \(3^{12}\cdot5^{2}\cdot11^{2}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((55)\) | = | \((5,a)^{2}\cdot(11,a)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 3025 \) | = | \(5^{2}\cdot11^{2}\) |
| j-invariant: | $j$ | = | \( \frac{59319}{55} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 3.4527523004098652830018694922289218553 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 13.81100920163946113200747796891568742 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 28.32995377530464766882549782925534260 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(1\cdot2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 7.6149896445357549648488428197312556072 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}7.614989645 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 14.164977 \cdot 13.811009 \cdot 4 } { {2^2 \cdot 25.690465} } \\ & \approx 7.614989645 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3,a)\) | \(3\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((5,a)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((11,a)\) | \(11\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
55.1-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 495.a4 |
| \(\Q\) | 48400.bt4 |