Base field \(\Q(\zeta_{28})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} + 14 x^{2} - 7 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{18}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(2 a^{5} + 2 a^{4} - 12 a^{3} - 12 a^{2} + 17 a + 17 : -9 a^{5} - 8 a^{4} + 56 a^{3} + 49 a^{2} - 83 a - 72 : 1\right)$ | $0.38968968522384656501173191464096630247$ | $\infty$ |
| $\left(2 a^{4} - 6 a^{2} + 3 : -8 a^{4} + 26 a^{2} - 16 : 1\right)$ | $0$ | $18$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((a^5-a^4-5a^3+6a^2+4a-7)\) | = | \((a^5-5a^3+5a)\cdot(-a^4+a^3+4a^2-3a-2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 56 \) | = | \(7\cdot8\) |
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| Discriminant: | $\Delta$ | = | $-28$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-28)\) | = | \((a^5-5a^3+5a)^{6}\cdot(-a^4+a^3+4a^2-3a-2)^{4}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 481890304 \) | = | \(7^{6}\cdot8^{4}\) |
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| j-invariant: | $j$ | = | \( -\frac{15625}{28} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 1 \) |
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| Mordell-Weil rank: | $r$ | = | \(1\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.38968968522384656501173191464096630247 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.33813811134307939007039148784579781482 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 44104.626108278665262787570673624561189 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 12 \) = \(( 2 \cdot 3 )\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(18\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.68261 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.682610000 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 44104.626108 \cdot 2.338138 \cdot 12 } { {18^2 \cdot 1037.134514} } \\ & \approx 3.682607679 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-5a^3+5a)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((-a^4+a^3+4a^2-3a-2)\) | \(8\) | \(2\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
56.1-a
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 5 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 14.a5 |
| \(\Q\) | 784.b5 |
| \(\Q(\sqrt{7}) \) | 2.2.28.1-14.1-b2 |
| \(\Q(\zeta_{7})^+\) | 3.3.49.1-56.1-a5 |
| \(\Q(\zeta_{7})^+\) | a curve with conductor norm 200704 (not in the database) |