Base field \(\Q(\sqrt{77}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 19 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(\frac{1665}{11} a + \frac{6671}{11} : \frac{633839}{121} a + \frac{2475986}{121} : 1\right)$ | $2.1152347890936445941256871048416713837$ | $\infty$ |
| $\left(\frac{9}{4} a + 10 : -\frac{31}{4} a - \frac{215}{8} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((6a+18)\) | = | \((2)\cdot(a+3)\cdot(3)\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 252 \) | = | \(4\cdot7\cdot9\) |
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| Discriminant: | $\Delta$ | = | $-4218578658$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-4218578658)\) | = | \((2)\cdot(a+3)^{4}\cdot(3)^{16}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 17796405893733080964 \) | = | \(4\cdot7^{4}\cdot9^{16}\) |
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| j-invariant: | $j$ | = | \( \frac{6359387729183}{4218578658} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 2.1152347890936445941256871048416713837 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 4.2304695781872891882513742096833427674 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 0.62172181914741884533070705368532153733 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 64 \) = \(1\cdot2^{2}\cdot2^{4}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 4.7957805882881140620646492046315154257 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}4.795780588 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 0.621722 \cdot 4.230470 \cdot 64 } { {2^2 \cdot 8.774964} } \\ & \approx 4.795780588 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((a+3)\) | \(7\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((3)\) | \(9\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
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
252.1-e
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 |
|---|---|
| \(\Q\) | 294.g6 |
| \(\Q\) | 5082.d6 |