Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+282300x-21683500\)
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(homogenize, simplify) |
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\(y^2z=x^3+282300xz^2-21683500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+282300x-21683500\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 412200 \) | = | $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $-1642949851500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{15} \cdot 5^{9} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{853235323904}{563425875} \) | = | $2^{10} \cdot 3^{-9} \cdot 5^{-3} \cdot 229^{-1} \cdot 941^{3}$ |
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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: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1840522971849908292576300368$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.36792907626058892331480633742$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8894533789406713$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8104087035216567$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.15179908800690320658623364477$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4287854081104513053797383164 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.428785408 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.151799 \cdot 1.000000 \cdot 16}{1^2} \\ & \approx 2.428785408\end{aligned}$$
Modular invariants
Modular form 412200.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4976640 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{1}^{*}$ | additive | -1 | 3 | 8 | 0 |
| $3$ | $2$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $229$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6870 = 2 \cdot 3 \cdot 5 \cdot 229 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 5731 & 2 \\ 5731 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6869 & 0 \end{array}\right),\left(\begin{array}{rr} 5497 & 2 \\ 5497 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2291 & 2 \\ 2291 & 3 \end{array}\right),\left(\begin{array}{rr} 6869 & 2 \\ 6868 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[6870])$ is a degree-$189250371993600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6870\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 51525 = 3^{2} \cdot 5^{2} \cdot 229 \) |
| $3$ | additive | $6$ | \( 45800 = 2^{3} \cdot 5^{2} \cdot 229 \) |
| $5$ | additive | $18$ | \( 16488 = 2^{3} \cdot 3^{2} \cdot 229 \) |
| $229$ | split multiplicative | $230$ | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 412200.bc consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 27480.k1, its twist by $-15$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.