Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-55232604x-157994289936\)
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(homogenize, simplify) |
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\(y^2z=x^3-55232604xz^2-157994289936z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-55232604x-157994289936\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8962, 258488)$ | $8.1056689188878453984916471753$ | $\infty$ |
Integral points
\((8962,\pm 258488)\)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $31604176116301824$ | = | $2^{14} \cdot 3^{9} \cdot 7^{8} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{10023392043504}{17} \) | = | $2^{4} \cdot 3^{6} \cdot 7^{4} \cdot 17^{-1} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8558104985192158171242597410$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.074093861338678182479180157355$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0483234584417949$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.976232080425791$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.1056689188878453984916471753$ |
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| Real period: | $\Omega$ | ≈ | $0.055379709716504487060865512739$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.5911167342888130625471394570 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.591116734 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.055380 \cdot 8.105669 \cdot 8}{1^2} \\ & \approx 3.591116734\end{aligned}$$
Modular invariants
Modular form 479808.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21288960 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $7$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $17$ | $1$ | $I_{1}$ | 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 \( 204 = 2^{2} \cdot 3 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 203 & 0 \end{array}\right),\left(\begin{array}{rr} 103 & 2 \\ 103 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 203 & 2 \\ 202 & 3 \end{array}\right),\left(\begin{array}{rr} 37 & 2 \\ 37 & 3 \end{array}\right),\left(\begin{array}{rr} 137 & 2 \\ 137 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[204])$ is a degree-$180486144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/204\Z)$.
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 | $2$ | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| $3$ | additive | $2$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $26$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479808v consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 29988r1, its twist by $-56$.
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.