Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-541622x+3612295496\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-541622xz^2+3612295496z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-701941491x+168537364497486\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(882, 61372)$ | $0.38162543809751122850117025260$ | $\infty$ |
Integral points
\( \left(-1554, 27268\right) \), \( \left(-1554, -25715\right) \), \( \left(882, 61372\right) \), \( \left(882, -62255\right) \), \( \left(19110, 2630932\right) \), \( \left(19110, -2650043\right) \)
Invariants
| Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5627001336330373429248$ | = | $-1 \cdot 2^{11} \cdot 3^{6} \cdot 7^{5} \cdot 13 \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{29540882258497}{9459954137088} \) | = | $-1 \cdot 2^{-11} \cdot 3^{-6} \cdot 7^{-5} \cdot 13^{-1} \cdot 19^{3} \cdot 29^{-1} \cdot 1627^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8526683596592290262554178405$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1690204446659920126637818243$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9653135148425609$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.413397212976726$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.38162543809751122850117025260$ |
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| Real period: | $\Omega$ | ≈ | $0.10993713785952114889851716219$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 120 $ = $ 1\cdot( 2 \cdot 3 )\cdot5\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0345770078591495547093101730 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.034577008 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.109937 \cdot 0.381625 \cdot 120}{1^2} \\ & \approx 5.034577008\end{aligned}$$
Modular invariants
Modular form 459186.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 31046400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 \( 21112 = 2^{3} \cdot 7 \cdot 13 \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 21111 & 0 \end{array}\right),\left(\begin{array}{rr} 10557 & 2 \\ 10557 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21111 & 2 \\ 21110 & 3 \end{array}\right),\left(\begin{array}{rr} 19489 & 2 \\ 19489 & 3 \end{array}\right),\left(\begin{array}{rr} 15081 & 2 \\ 15081 & 3 \end{array}\right),\left(\begin{array}{rr} 5279 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7281 & 2 \\ 7281 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[21112])$ is a degree-$27677122961080320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21112\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 76531 = 7 \cdot 13 \cdot 29^{2} \) |
| $3$ | split multiplicative | $4$ | \( 153062 = 2 \cdot 7 \cdot 13 \cdot 29^{2} \) |
| $5$ | good | $2$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
| $7$ | split multiplicative | $8$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
| $11$ | good | $2$ | \( 229593 = 3 \cdot 7 \cdot 13 \cdot 29^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 35322 = 2 \cdot 3 \cdot 7 \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 459186bc consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 15834o1, its twist by $29$.
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.