Base field \(\Q(\sqrt{65}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 16 \); class number \(2\).
Weierstrass equation
This is not a global minimal model: it is minimal at all primes except \((2,a)\). No global minimal model exists.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-a + 7 : 13 a - 72 : 1\right)$ | $1.6691378484290617787436997951507660342$ | $\infty$ |
| $\left(-a + 3 : -a + 8 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-5a+20)\) | = | \((2,a)^{2}\cdot(5,a+2)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 100 \) | = | \(2^{2}\cdot5^{2}\) |
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| Discriminant: | $\Delta$ | = | $190625a+330000$ | ||
| Discriminant ideal: | $(\Delta)$ | = | \((190625a+330000)\) | = | \((2,a)^{20}\cdot(5,a+2)^{8}\) |
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| Discriminant norm: | $N(\Delta)$ | = | \( -409600000000 \) | = | \(-2^{20}\cdot5^{8}\) |
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| Minimal discriminant: | $\frak{D}_{\mathrm{min}}$ | = | \((625a+10000)\) | = | \((2,a)^{8}\cdot(5,a+2)^{8}\) |
| Minimal discriminant norm: | $N(\frak{D}_{\mathrm{min}})$ | = | \( -100000000 \) | = | \(-2^{8}\cdot5^{8}\) |
| j-invariant: | $j$ | = | \( \frac{2941}{5} a + \frac{18368}{5} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.6691378484290617787436997951507660342 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 3.3382756968581235574873995903015320684 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 6.6693304288843908392419685480461369784 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 6 \) = \(3\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.1422758450806870452331461152533165865 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.142275845 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 6.669330 \cdot 3.338276 \cdot 6 } { {2^2 \cdot 8.062258} } \\ & \approx 4.142275845 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $\frak{p}$ 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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((5,a+2)\) | \(5\) | \(2\) | \(I_{2}^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(2\) |
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
100.2-b
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.