Base field \(\Q(\sqrt{-14}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 14 \); class number \(4\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(871 : 25687 : 1\right)$ | $6.7911921286310453797986406061281382199$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((27)\) | = | \((3,a+1)^{3}\cdot(3,a+2)^{3}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 729 \) | = | \(3^{3}\cdot3^{3}\) |
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| Discriminant: | $\Delta$ | = | $11337408$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((11337408)\) | = | \((2,a)^{12}\cdot(3,a+1)^{11}\cdot(3,a+2)^{11}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 128536820158464 \) | = | \(2^{12}\cdot3^{11}\cdot3^{11}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((177147)\) | = | \((3,a+1)^{11}\cdot(3,a+2)^{11}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 31381059609 \) | = | \(3^{11}\cdot3^{11}\) |
| j-invariant: | $j$ | = | \( -12288000 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z\) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-27})/2]\) (potential complex multiplication) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 6.7911921286310453797986406061281382199 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 13.582384257262090759597281212256276440 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.80191739201393214241028956008613560804 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.2705205057909614028568413128931658483 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.270520506 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.801917 \cdot 13.582384 \cdot 1 } { {1^2 \cdot 7.483315} } \\ & \approx 3.270520506 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
| \((3,a+2)\) | \(3\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(3\) | \(11\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cs |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 9 and 27.
Its isogeny class
729.4-c
consists of curves linked by isogenies of
degrees dividing 27.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 1323.i1 |
| \(\Q\) | 1728.n1 |