Base field \(\Q(\sqrt{-2491}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 623 \); class number \(12\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((5,a+1)\). No global minimal model exists.
Mordell-Weil group structure
Not computed ($ 0 \le r \le 1 $)
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 4 \) | = | \(4\) |
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| Discriminant: | $\Delta$ | = | $80740352a+326500352$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((80740352a+326500352)\) | = | \((2)^{17}\cdot(5,a+1)^{12}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( 4194304000000000000 \) | = | \(4^{17}\cdot5^{12}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((131072)\) | = | \((2)^{17}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( 17179869184 \) | = | \(4^{17}\) |
| j-invariant: | $j$ | = | \( -\frac{25153757}{131072} \) | ||
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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: | $\Omega(E/K)$ | ≈ | \( 7.7982180460166276079094734025084930424 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) = \(1\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.6264591614019008371606670768193544162 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | not available |
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
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| \((2)\) | \(4\) | \(1\) | \(I_{17}\) | Non-split multiplicative | \(1\) | \(1\) | \(17\) | \(17\) |
| \((5,a+1)\) | \(5\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 4.1-c consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.