Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); 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 |
|---|---|---|
| $\left(-28 i : 54 i + 26 : 1\right)$ | $1.4975362857218438153786031768718604524$ | $\infty$ |
| $\left(\frac{25}{2} i : -\frac{27}{4} i + \frac{23}{4} : 1\right)$ | $0$ | $2$ |
Invariants
| Conductor: | $\frak{N}$ | = | \((90i+90)\) | = | \((i+1)^{3}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}\) |
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| Conductor norm: | $N(\frak{N})$ | = | \( 16200 \) | = | \(2^{3}\cdot5\cdot5\cdot9^{2}\) |
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| Discriminant: | $\Delta$ | = | $-9447840$ | ||
| Discriminant ideal: | $\frak{D}_{\mathrm{min}} = (\Delta)$ | = | \((-9447840)\) | = | \((i+1)^{10}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{10}\) |
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| Discriminant norm: | $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ | = | \( 89261680665600 \) | = | \(2^{10}\cdot5\cdot5\cdot9^{10}\) |
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| j-invariant: | $j$ | = | \( \frac{546718898}{405} \) | ||
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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: | $\mathrm{Reg}(E/K)$ | ≈ | \( 1.4975362857218438153786031768718604524 \) |
| Néron-Tate Regulator: | $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ | ≈ | \( 2.9950725714436876307572063537437209048 \) |
| Global period: | $\Omega(E/K)$ | ≈ | \( 1.16052896191482170303194636741208813582 \) |
| Tamagawa product: | $\prod_{\frak{p}}c_{\frak{p}}$ | = | \( 8 \) = \(2\cdot1\cdot1\cdot2^{2}\) |
| Torsion order: | $\#E(K)_{\mathrm{tor}}$ | = | \(2\) |
| Special value: | $L^{(r)}(E/K,1)/r!$ | ≈ | \( 3.4758684621970984666435975087353492155 \) |
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | \( 1 \) (rounded) |
BSD formula
$$\begin{aligned}3.475868462 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.160529 \cdot 2.995073 \cdot 8 } { {2^2 \cdot 2.000000} } \\ & \approx 3.475868462 \end{aligned}$$
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 | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) | \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(3\) | \(10\) | \(0\) |
| \((-i-2)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((2i+1)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((3)\) | \(9\) | \(4\) | \(I_{4}^{*}\) | Additive | \(1\) | \(2\) | \(10\) | \(4\) |
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 and 4.
Its isogeny class
16200.2-d
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 |
|---|---|
| \(\Q\) | 360.e1 |
| \(\Q\) | 720.f1 |