Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1383123x-3503386222\)
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(homogenize, simplify) |
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\(y^2z=x^3-1383123xz^2-3503386222z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1383123x-3503386222\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(42682199/1681, 278520546710/68921)$ | $14.164781254656884399453589264$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 458640 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-5132903786006917939200$ | = | $-1 \cdot 2^{29} \cdot 3^{6} \cdot 5^{2} \cdot 7^{9} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{832972004929}{14611251200} \) | = | $-1 \cdot 2^{-17} \cdot 5^{-2} \cdot 7^{-3} \cdot 13^{-1} \cdot 97^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8469710079495221529303615842$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.63156260852786534526283047256$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1059450258035821$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.409101349322575$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $14.164781254656884399453589264$ |
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| Real period: | $\Omega$ | ≈ | $0.058600143450676792523320763181$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6404657077628081950853864884 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.640465708 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.058600 \cdot 14.164781 \cdot 8}{1^2} \\ & \approx 6.640465708\end{aligned}$$
Modular invariants
Modular form 458640.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 28200960 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{21}^{*}$ | additive | -1 | 4 | 29 | 17 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $13$ | $1$ | $I_{1}$ | 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 \( 728 = 2^{3} \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 183 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 2 \\ 561 & 3 \end{array}\right),\left(\begin{array}{rr} 521 & 2 \\ 521 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 727 & 2 \\ 726 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 727 & 0 \end{array}\right),\left(\begin{array}{rr} 365 & 2 \\ 365 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[728])$ is a degree-$40577531904$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.
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 | $4$ | \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 50960 = 2^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 91728 = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $32$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 35280 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 458640p consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 910h1, its twist by $-84$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.