Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+1555933x+6772122341\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+1555933xz^2+6772122341z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+24894925x+433440724750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-5594/9, 2022491/27)$ | $6.6967906540263458916637858940$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 477950 \) | = | $2 \cdot 5^{2} \cdot 11^{2} \cdot 79$ |
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| Discriminant: | $\Delta$ | = | $-20055580139160156250000$ | = | $-1 \cdot 2^{4} \cdot 5^{23} \cdot 11^{3} \cdot 79 $ |
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| j-invariant: | $j$ | = | \( \frac{20030793296215941}{964355468750000} \) | = | $2^{-4} \cdot 3^{3} \cdot 5^{-17} \cdot 79^{-1} \cdot 90527^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9591447941893349749358529405$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5549520197726921516199873794$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0158080752002754$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.49600411978685$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.6967906540263458916637858940$ |
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| Real period: | $\Omega$ | ≈ | $0.092329893400992081771164830505$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.9465117377201004872160947290 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.946511738 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.092330 \cdot 6.696791 \cdot 8}{1^2} \\ & \approx 4.946511738\end{aligned}$$
Modular invariants
Modular form 477950.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 26947584 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_{17}^{*}$ | additive | 1 | 2 | 23 | 17 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $79$ | $1$ | $I_{1}$ | nonsplit 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 \( 17380 = 2^{2} \cdot 5 \cdot 11 \cdot 79 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 12641 & 2 \\ 12641 & 3 \end{array}\right),\left(\begin{array}{rr} 8691 & 2 \\ 8691 & 3 \end{array}\right),\left(\begin{array}{rr} 11221 & 2 \\ 11221 & 3 \end{array}\right),\left(\begin{array}{rr} 17379 & 2 \\ 17378 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 17379 & 0 \end{array}\right),\left(\begin{array}{rr} 3477 & 2 \\ 3477 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[17380])$ is a degree-$11693989232640000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17380\Z)$.
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 | $4$ | \( 21725 = 5^{2} \cdot 11 \cdot 79 \) |
| $5$ | additive | $18$ | \( 19118 = 2 \cdot 11^{2} \cdot 79 \) |
| $11$ | additive | $42$ | \( 3950 = 2 \cdot 5^{2} \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 477950v consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 95590l1, its twist by $5$.
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.