Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-22163x-3835965\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-22163xz^2-3835965z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-354603x-245856346\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2415/4, 111447/8)$ | $5.1252159365063761314176239359$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 431766 \) | = | $2 \cdot 3^{2} \cdot 17^{2} \cdot 83$ |
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| Discriminant: | $\Delta$ | = | $-5678392458533904$ | = | $-1 \cdot 2^{4} \cdot 3^{11} \cdot 17^{6} \cdot 83 $ |
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| j-invariant: | $j$ | = | \( -\frac{68417929}{322704} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-5} \cdot 83^{-1} \cdot 409^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7075880821731801586227739845$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.25832473418898272719961594290$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9061889513754705$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.379093323806923$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1252159365063761314176239359$ |
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| Real period: | $\Omega$ | ≈ | $0.17713048813245462022428589442$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.525312009881759195311450849 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.525312010 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.177130 \cdot 5.125216 \cdot 16}{1^2} \\ & \approx 14.525312010\end{aligned}$$
Modular invariants
Modular form 431766.2.a.bp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3153920 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{5}^{*}$ | additive | -1 | 2 | 11 | 5 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $83$ | $1$ | $I_{1}$ | 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 \( 996 = 2^{2} \cdot 3 \cdot 83 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 995 & 2 \\ 994 & 3 \end{array}\right),\left(\begin{array}{rr} 665 & 2 \\ 665 & 3 \end{array}\right),\left(\begin{array}{rr} 499 & 2 \\ 499 & 3 \end{array}\right),\left(\begin{array}{rr} 85 & 2 \\ 85 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 995 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[996])$ is a degree-$108010893312$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/996\Z)$.
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 | $4$ | \( 215883 = 3^{2} \cdot 17^{2} \cdot 83 \) |
| $3$ | additive | $8$ | \( 47974 = 2 \cdot 17^{2} \cdot 83 \) |
| $17$ | additive | $146$ | \( 1494 = 2 \cdot 3^{2} \cdot 83 \) |
| $83$ | nonsplit multiplicative | $84$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 431766.bp consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 498.a1, its twist by $-51$.
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.