Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-578625x+357041525\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-578625xz^2+357041525z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-749898675x+16669377867150\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-955, 6785)$ | $2.8700103548588503877951346736$ | $\infty$ |
Integral points
\( \left(-955, 6785\right) \), \( \left(-955, -5830\right) \)
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-42746573483806720000$ | = | $-1 \cdot 2^{15} \cdot 5^{4} \cdot 11^{2} \cdot 29^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{57629979025}{114982912} \) | = | $-1 \cdot 2^{-15} \cdot 5^{2} \cdot 11^{-2} \cdot 29^{-1} \cdot 1321^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4562377112599559939283052642$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.23611049212201885546974947028$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8908792280729944$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.056206108203606$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8700103548588503877951346736$ |
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| Real period: | $\Omega$ | ≈ | $0.18084478041268525983689749657$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot3\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.2283167088789803815588425236 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.228316709 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.180845 \cdot 2.870010 \cdot 12}{1^2} \\ & \approx 6.228316709\end{aligned}$$
Modular invariants
Modular form 462550.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20563200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{15}$ | nonsplit multiplicative | 1 | 1 | 15 | 15 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $29$ | $2$ | $I_{1}^{*}$ | 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 \( 232 = 2^{3} \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 89 & 2 \\ 89 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 231 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 117 & 2 \\ 117 & 3 \end{array}\right),\left(\begin{array}{rr} 231 & 2 \\ 230 & 3 \end{array}\right),\left(\begin{array}{rr} 175 & 2 \\ 175 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[232])$ is a degree-$523837440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/232\Z)$.
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 | $4$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $3$ | good | $2$ | \( 231275 = 5^{2} \cdot 11 \cdot 29^{2} \) |
| $5$ | additive | $14$ | \( 9251 = 11 \cdot 29^{2} \) |
| $11$ | split multiplicative | $12$ | \( 42050 = 2 \cdot 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462550.be consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 15950.k1, 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.