Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-402739403x-3111079747395\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-402739403xz^2-3111079747395z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-521950266963x-145142707440460050\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9550616124683680340488829358053275991732896812736706834279232964427926130016713135923287858675813115/287921578744348452954232456417354794736338837545787871579239898683212414623162761533627068501369, 688860851995986696247999383644007790573255239481084105344087808969329210948440266642499408398457873915431227011951544363565060354617153954580431963945/154493895009282732142114150032972393240824245853449953566449878665048152719969574701743667774579219646304567932866064847838726529701383733355597)$ | $226.66197537063382928856005411$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 490398 \) | = | $2 \cdot 3 \cdot 37 \cdot 47^{2}$ |
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| Discriminant: | $\Delta$ | = | $-65852166022028876316672$ | = | $-1 \cdot 2^{23} \cdot 3^{9} \cdot 37 \cdot 47^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{670206957616537490521}{6109179936768} \) | = | $-1 \cdot 2^{-23} \cdot 3^{-9} \cdot 37^{-1} \cdot 2477^{3} \cdot 3533^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5430673309248436657920850414$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6179935300698143723816097065$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0959689233554826$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.42281677913196$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $226.66197537063382928856005411$ |
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| Real period: | $\Omega$ | ≈ | $0.016850519352086785408509405180$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6387440047301673301176695531 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.638744005 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.016851 \cdot 226.661975 \cdot 2}{1^2} \\ & \approx 7.638744005\end{aligned}$$
Modular invariants
Modular form 490398.2.a.f
For more coefficients, see the Downloads section to the right.
| Modular degree: | 251837856 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{23}$ | nonsplit multiplicative | 1 | 1 | 23 | 23 |
| $3$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $37$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $47$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 \( 888 = 2^{3} \cdot 3 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 887 & 0 \end{array}\right),\left(\begin{array}{rr} 223 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 409 & 2 \\ 409 & 3 \end{array}\right),\left(\begin{array}{rr} 887 & 2 \\ 886 & 3 \end{array}\right),\left(\begin{array}{rr} 593 & 2 \\ 593 & 3 \end{array}\right),\left(\begin{array}{rr} 445 & 2 \\ 445 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[888])$ is a degree-$67172696064$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/888\Z)$.
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 | $4$ | \( 245199 = 3 \cdot 37 \cdot 47^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 163466 = 2 \cdot 37 \cdot 47^{2} \) |
| $23$ | good | $2$ | \( 245199 = 3 \cdot 37 \cdot 47^{2} \) |
| $37$ | split multiplicative | $38$ | \( 13254 = 2 \cdot 3 \cdot 47^{2} \) |
| $47$ | additive | $1106$ | \( 222 = 2 \cdot 3 \cdot 37 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 490398.f consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 222.a1, its twist by $-47$.
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.