Base field 4.4.17069.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} - 4 x + 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{5}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-\frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{10}{9} a + \frac{5}{12} : -\frac{17}{54} a^{3} + \frac{11}{18} a^{2} + \frac{40}{27} a - \frac{41}{72} : 1\right)$ | $2.0274512565601798892765309541460133694$ | $\infty$ |
| $\left(1 : 0 : 1\right)$ | $0$ | $5$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((2)\) | = | \((2)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16 \) | = | \(16\) |
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| Discriminant: | $\Delta$ | = | $-2$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-2)\) | = | \((2)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 16 \) | = | \(16\) |
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| j-invariant: | $j$ | = | \( \frac{461373}{2} a^{3} - 461373 a^{2} - \frac{2306865}{2} a - \frac{601423}{2} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.0274512565601798892765309541460133694 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 8.1098050262407195571061238165840534776 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 815.47446296687401905343501797177085661 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 1 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(5\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.02477481707981 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.024774817 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 815.474463 \cdot 8.109805 \cdot 1 } { {5^2 \cdot 130.648383} } \\ & \approx 2.024774817 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(16\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 |
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5, 9, 15 and 45.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 45.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{13}) \) | 2.2.13.1-4.1-a4 |
| \(\Q(\sqrt{13}) \) | a curve with conductor norm 40804 (not in the database) |