Base field \(\Q(\sqrt{-111}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 28 \); class number \(8\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((7,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(8 a - 24 : -15 a + 26 : 1\right)$ | $0.21949447891908335890241016544986685664$ | $\infty$ |
| $\left(6 a - \frac{100}{3} : \frac{161}{9} a - \frac{214}{9} : 1\right)$ | $0$ | $3$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((10,2a+2)\) | = | \((2,a)\cdot(2,a+1)\cdot(5,a+1)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 20 \) | = | \(2\cdot2\cdot5\) |
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| Discriminant: | $\Delta$ | = | $-23004107264a+106865342976$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((-23004107264a+106865342976)\) | = | \((2,a)^{9}\cdot(2,a+1)^{27}\cdot(5,a+1)^{2}\cdot(7,a)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 23779150345135351398400 \) | = | \(2^{9}\cdot2^{27}\cdot5^{2}\cdot7^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((-189952a+941568)\) | = | \((2,a)^{9}\cdot(2,a+1)^{27}\cdot(5,a+1)^{2}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 1717986918400 \) | = | \(2^{9}\cdot2^{27}\cdot5^{2}\) |
| j-invariant: | $j$ | = | \( \frac{87441895536793}{3355443200} a + \frac{60246101822861}{838860800} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.21949447891908335890241016544986685664 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.438988957838166717804820330899733713280 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.95256283459812827154612149305478795950 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 486 \) = \(3^{2}\cdot3^{3}\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(3\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.3933002521699794022595753946915991204 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.393300252 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.952563 \cdot 0.438989 \cdot 486 } { {3^2 \cdot 10.535654} } \\ & \approx 4.393300252 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((2,a+1)\) | \(2\) | \(27\) | \(I_{27}\) | Split multiplicative | \(-1\) | \(1\) | \(27\) | \(27\) |
| \((5,a+1)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((7,a)\) | \(7\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
20.3-b
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.