Base field \(\Q(\sqrt{-19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 5 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $\left(0 : 9 : 1\right)$ | $0.18129167403579927415788154138376329986$ | $\infty$ |
| $\left(-\frac{4}{25} a + \frac{59}{25} : -\frac{57}{125} a - \frac{528}{125} : 1\right)$ | $1.2690417182505949191051707896863430990$ | $\infty$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((19)\) | = | \((-2a+1)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 361 \) | = | \(19^{2}\) |
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| Discriminant: | $\Delta$ | = | $6859$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((6859)\) | = | \((-2a+1)^{6}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 47045881 \) | = | \(19^{6}\) |
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| j-invariant: | $j$ | = | \( -884736 \) | ||
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | \(\Z[(1+\sqrt{-19})/2]\) (complex multiplication) | ||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-19})/2]\) | ||
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{U}(1)$ | ||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | \( 2 \) |
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| Mordell-Weil rank: | $r$ | = | \(2\) |
| Regulator: | $\mathrm{Reg}(E/K)$ | ≈ | \( 0.15611668760483685923672674998964392243 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 0.624466750419347436946906999958575689720 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 4.0287030500025882741147230622254799474 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 4 \) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(1\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 2.3086482477578043057816112437958900992 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}2.308648248 \approx L^{(2)}(E/K,1)/2! & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 4.028703 \cdot 0.624467 \cdot 4 } { {1^2 \cdot 4.358899} } \\ & \approx 2.308648248 \end{aligned}$$
Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.
| $\mathfrak{p}$ | $N(\mathfrak{p})$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(19\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(19\) | 19B.1.17[2] |
For all other primes \(p\), the image is a split Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 361.1-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 361.a1 |
| \(\Q\) | 361.a2 |