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 \((2,a+1)\). No global minimal model exists.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 1 $)
Invariants
| Conductor: | $\frak{N}$ | = | \((147,3a+81)\) | = | \((3,a+1)^{2}\cdot(7,a+6)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 441 \) | = | \(3^{2}\cdot7^{2}\) |
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| Discriminant: | $\Delta$ | = | $60507a+469449$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((60507a+469449)\) | = | \((2,a+1)^{12}\cdot(3,a+1)^{6}\cdot(7,a+6)^{6}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 351298031616 \) | = | \(2^{12}\cdot3^{6}\cdot7^{6}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((3176523,27a+1596267)\) | = | \((3,a+1)^{6}\cdot(7,a+6)^{6}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 85766121 \) | = | \(3^{6}\cdot7^{6}\) |
| j-invariant: | $j$ | = | \( 38477541376 \) | ||
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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: | $r?$ | \(0 \le r \le 1\) | |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | not available |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | not available |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2.3006155519362630481848245562679215678 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 26.484319547230476178164771260448614476 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | not available |
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a+1)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((7,a+6)\) | \(7\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Cn |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
441.3-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.