Base field \(\Q(\sqrt{13}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(-3 : 14 : 1\right)$ | $0.58523207679736111555815040016259481988$ | $\infty$ |
| $\left(10 : 27 : 1\right)$ | $0$ | $6$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((-20a+10)\) | = | \((2)\cdot(-2a+1)\cdot(5)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 1300 \) | = | \(4\cdot13\cdot25\) |
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| Discriminant: | $\Delta$ | = | $-10562500$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-10562500)\) | = | \((2)^{2}\cdot(-2a+1)^{4}\cdot(5)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 111566406250000 \) | = | \(4^{2}\cdot13^{4}\cdot25^{6}\) |
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| j-invariant: | $j$ | = | \( -\frac{217081801}{10562500} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.58523207679736111555815040016259481988 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 1.17046415359472223111630080032518963976 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 3.5800972799118437201846604851385217188 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 48 \) = \(2\cdot2^{2}\cdot( 2 \cdot 3 )\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(6\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 1.5496014200970454616139485549050371674 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}1.549601420 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.580097 \cdot 1.170464 \cdot 48 } { {6^2 \cdot 3.605551} } \\ & \approx 1.549601420 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((-2a+1)\) | \(13\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((5)\) | \(25\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
1300.1-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 130.a3 |
| \(\Q\) | 1690.f3 |