Base field \(\Q(\sqrt{3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(a - 3 : 2 a - 3 : 1\right)$ | $0.29482882409632301002610224036666681076$ | $\infty$ |
| $\left(\frac{3}{2} a - 4 : \frac{3}{4} a - \frac{3}{4} : 1\right)$ | $0$ | $2$ |
| $\left(-3 a + 4 : -a + 2 : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((26a)\) | = | \((a+1)^{2}\cdot(a)\cdot(a+4)\cdot(a-4)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 2028 \) | = | \(2^{2}\cdot3\cdot13\cdot13\) |
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| Discriminant: | $\Delta$ | = | $2028$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((2028)\) | = | \((a+1)^{4}\cdot(a)^{2}\cdot(a+4)^{2}\cdot(a-4)^{2}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 4112784 \) | = | \(2^{4}\cdot3^{2}\cdot13^{2}\cdot13^{2}\) |
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| j-invariant: | $j$ | = | \( \frac{3631696}{507} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.29482882409632301002610224036666681076 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.589657648192646020052204480733333621520 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 10.297812852922467793550350978307822292 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 24 \) = \(3\cdot2\cdot2\cdot2\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(4\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.6293328471576012817677808826908610029 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.629332847 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 10.297813 \cdot 0.589658 \cdot 24 } { {4^2 \cdot 3.464102} } \\ & \approx 2.629332847 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(3\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
| \((a)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((a+4)\) | \(13\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((a-4)\) | \(13\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
2028.1-g
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 156.a1 |
| \(\Q\) | 1872.s1 |