Base field 4.4.12725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 10 x^{2} + 11 x + 29 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(12 a^{3} - 42 a^{2} - 49 a + 200 : 115 a^{3} - 424 a^{2} - 453 a + 2042 : 1\right)$ | $0.078203491736569322144030942907611869668$ | $\infty$ |
| $\left(\frac{349072}{116281} a^{3} - \frac{119018}{10571} a^{2} - \frac{1405483}{116281} a + \frac{6260404}{116281} : \frac{29025661}{39651821} a^{3} - \frac{20092572}{3604711} a^{2} - \frac{8587959}{39651821} a + \frac{1169807946}{39651821} : 1\right)$ | $2.8935291942530649193291448875816391777$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16 \) | = | \(16\) |
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| Discriminant: | $\Delta$ | = | $-4$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-4)\) | = | \((2)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 256 \) | = | \(16^{2}\) |
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| j-invariant: | $j$ | = | \( -\frac{6297835481}{4} a^{2} + \frac{6297835481}{4} a + \frac{27631544681}{4} \) | ||
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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: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.0061157861197916662528680154185111093091 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.097852577916666660045888246696177748945600 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 2223.9649180752091523008307397569128194 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 2 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.85834718344607 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.858347183 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2223.964918 \cdot 0.097853 \cdot 2 } { {1^2 \cdot 112.805142} } \\ & \approx 3.858347183 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(16\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 16.1-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.