Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-484x+1331\)
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(homogenize, simplify) |
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\(y^2z=x^3-484xz^2+1331z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-484x+1331\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1295/324, 329993/5832)$ | $8.9978611858713865865398546799$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 443344 \) | = | $2^{4} \cdot 11^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $6490999504$ | = | $2^{4} \cdot 11^{6} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{442368}{229} \) | = | $2^{14} \cdot 3^{3} \cdot 229^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.57482054981932480782238607446$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.85517614676650890068099642168$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7674474656626996$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.3196141241597585$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.9978611858713865865398546799$ |
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| Real period: | $\Omega$ | ≈ | $1.1765351290267688889484281053$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.586299771284147541861500355 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.586299771 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.176535 \cdot 8.997861 \cdot 1}{1^2} \\ & \approx 10.586299771\end{aligned}$$
Modular invariants
Modular form 443344.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 712800 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 4 | 4 | 0 |
| $11$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $229$ | $1$ | $I_{1}$ | 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 \( 458 = 2 \cdot 229 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 457 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 235 & 2 \\ 235 & 3 \end{array}\right),\left(\begin{array}{rr} 457 & 2 \\ 456 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[458])$ is a degree-$8213991840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/458\Z)$.
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 | $2$ | \( 27709 = 11^{2} \cdot 229 \) |
| $11$ | additive | $62$ | \( 3664 = 2^{4} \cdot 229 \) |
| $229$ | nonsplit multiplicative | $230$ | \( 1936 = 2^{4} \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 443344p consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 916c1, its twist by $44$.
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.