Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-36161707x+83692065743\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-36161707xz^2+83692065743z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-46865572299x+3904877616022278\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(13967/4, -13967/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 494214 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 41^{2}$ |
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| Discriminant: | $\Delta$ | = | $300659735141009740818$ | = | $2 \cdot 3^{8} \cdot 7^{6} \cdot 41^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{9357915116017}{538002} \) | = | $2^{-1} \cdot 3^{-8} \cdot 13^{3} \cdot 41^{-1} \cdot 1621^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9928832590526520104215001368$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.16314215117284145593544207856$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9826480023940802$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.868086416533526$ | |||
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: | $\Omega$ | ≈ | $0.16335626704865176923406960180$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{3}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $11.761651227502927384853011330 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 11.761651228 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.163356 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 11.761651228\end{aligned}$$
Modular invariants
Modular form 494214.2.a.ef
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46448640 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $41$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2296 = 2^{3} \cdot 7 \cdot 41 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2289 & 8 \\ 2288 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2290 & 2291 \end{array}\right),\left(\begin{array}{rr} 288 & 1771 \\ 2177 & 2206 \end{array}\right),\left(\begin{array}{rr} 1275 & 616 \\ 2114 & 1933 \end{array}\right),\left(\begin{array}{rr} 468 & 1967 \\ 1953 & 650 \end{array}\right),\left(\begin{array}{rr} 655 & 0 \\ 0 & 2295 \end{array}\right)$.
The torsion field $K:=\Q(E[2296])$ is a degree-$177743462400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2296\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 82369 = 7^{2} \cdot 41^{2} \) |
| $3$ | split multiplicative | $4$ | \( 164738 = 2 \cdot 7^{2} \cdot 41^{2} \) |
| $7$ | additive | $26$ | \( 10086 = 2 \cdot 3 \cdot 41^{2} \) |
| $41$ | additive | $882$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 494214.ef
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 246.f1, its twist by $-287$.
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 $0$.