Base field \(\Q(\sqrt{17}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{5}{4} a + 7 : \frac{45}{8} a + \frac{17}{2} : 1\right)$ | $1.2700136100045908554936356119484536169$ | $\infty$ |
| $\left(6 : -4 : 1\right)$ | $0$ | $4$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((24)\) | = | \((-a+2)^{3}\cdot(-a-1)^{3}\cdot(3)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 576 \) | = | \(2^{3}\cdot2^{3}\cdot9\) |
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| Discriminant: | $\Delta$ | = | $3072$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((3072)\) | = | \((-a+2)^{10}\cdot(-a-1)^{10}\cdot(3)\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 9437184 \) | = | \(2^{10}\cdot2^{10}\cdot9\) |
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| j-invariant: | $j$ | = | \( \frac{28756228}{3} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.2700136100045908554936356119484536169 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.5400272200091817109872712238969072338 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 18.602238951643222078128233801763314336 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) = \(2\cdot2\cdot1\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.8649637906869380618902001105138190271 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.864963791 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 18.602239 \cdot 2.540027 \cdot 4 } { {4^2 \cdot 4.123106} } \\ & \approx 2.864963791 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((-a-1)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(-1\) | \(3\) | \(10\) | \(0\) |
| \((3)\) | \(9\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
576.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 24.a2 |
| \(\Q\) | 6936.p2 |