Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-554701x+159007048\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-554701xz^2+159007048z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-718891875x+7420789518750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 462550 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5365580000000000$ | = | $-1 \cdot 2^{11} \cdot 5^{10} \cdot 11 \cdot 29^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{79251117725}{22528} \) | = | $-1 \cdot 2^{-11} \cdot 5^{2} \cdot 11^{-1} \cdot 13^{3} \cdot 113^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9989527613110145560923740337$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.18406945654735426287074341874$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9184984529586074$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9321251080387025$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.41967604283777205154600162845$ |
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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)$ | ≈ | $3.3574083427021764123680130276 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.357408343 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.419676 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 3.357408343\end{aligned}$$
Modular invariants
Modular form 462550.2.a.bb
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6283200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{11}$ | nonsplit multiplicative | 1 | 1 | 11 | 11 |
| $5$ | $1$ | $II^{*}$ | additive | 1 | 2 | 10 | 0 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 \( 2552 = 2^{3} \cdot 11 \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 233 & 2 \\ 233 & 3 \end{array}\right),\left(\begin{array}{rr} 2551 & 2 \\ 2550 & 3 \end{array}\right),\left(\begin{array}{rr} 639 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1277 & 2 \\ 1277 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 89 & 2 \\ 89 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2551 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[2552])$ is a degree-$6914654208000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2552\Z)$.
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 | $4$ | \( 7975 = 5^{2} \cdot 11 \cdot 29 \) |
| $5$ | additive | $2$ | \( 18502 = 2 \cdot 11 \cdot 29^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 21025 = 5^{2} \cdot 29^{2} \) |
| $29$ | additive | $226$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 462550.bb consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 462550.bm1, its twist by $5$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.