Base field 4.4.13525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 8 x + 29 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{22}{5} a^{3} - 8 a^{2} - \frac{184}{5} a + \frac{347}{5} : -\frac{97}{5} a^{3} + 48 a^{2} + \frac{834}{5} a - \frac{1977}{5} : 1\right)$ | $0.27197669492073823381465950777209713380$ | $\infty$ |
| $\left(\frac{13}{5} a^{3} - 5 a^{2} - \frac{111}{5} a + \frac{847}{20} : -\frac{13}{10} a^{3} + 2 a^{2} + \frac{111}{10} a - \frac{747}{40} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((3/5a^3-16/5a-2/5)\) | = | \((3/5a^3+a^2-16/5a-22/5)\cdot(2/5a^3+a^2-14/5a-28/5)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 55 \) | = | \(5\cdot11\) |
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| Discriminant: | $\Delta$ | = | $-11229/5a^3-9508a^2+101833/5a+263731/5$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-11229/5a^3-9508a^2+101833/5a+263731/5)\) | = | \((3/5a^3+a^2-16/5a-22/5)^{22}\cdot(2/5a^3+a^2-14/5a-28/5)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( -288486480712890625 \) | = | \(-5^{22}\cdot11^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{13411008321949}{5908203125} a^{3} - \frac{48358554576758}{5908203125} a^{2} + \frac{24069628976738}{5908203125} a + \frac{416249632496594}{5908203125} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.27197669492073823381465950777209713380 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.08790677968295293525863803108838853520 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 39.895510529011602984067344997988152083 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 44 \) = \(( 2 \cdot 11 )\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.10525140551891 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.105251406 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 39.895511 \cdot 1.087907 \cdot 44 } { {2^2 \cdot 116.297033} } \\ & \approx 4.105251406 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((3/5a^3+a^2-16/5a-22/5)\) | \(5\) | \(22\) | \(I_{22}\) | Split multiplicative | \(-1\) | \(1\) | \(22\) | \(22\) |
| \((2/5a^3+a^2-14/5a-28/5)\) | \(11\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
55.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.