Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+51933644x+462858872779\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+51933644xz^2+462858872779z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+67306002192x+21594335896360656\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1789891/9604, 633592756385/941192)$ | $12.667936306500064129996816677$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 429429 \) | = | $3 \cdot 7 \cdot 11^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-101508544257831509795615619$ | = | $-1 \cdot 3^{8} \cdot 7^{7} \cdot 11^{6} \cdot 13^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{1811564780171264}{11870974573731} \) | = | $2^{12} \cdot 3^{-8} \cdot 7^{-7} \cdot 13^{-3} \cdot 19^{3} \cdot 401^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.6719330255260518099717766088$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1905107103960981699140610990$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0472072211574832$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.184529625863033$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.667936306500064129996816677$ |
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| Real period: | $\Omega$ | ≈ | $0.043363167370584600018957978990$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{3}\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $17.578298953557362418871161823 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 17.578298954 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043363 \cdot 12.667936 \cdot 32}{1^2} \\ & \approx 17.578298954\end{aligned}$$
Modular invariants
Modular form 429429.2.a.dv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 158054400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 \( 182 = 2 \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 181 & 2 \\ 180 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 181 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 15 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[182])$ is a degree-$158505984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/182\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 143143 = 7 \cdot 11^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 143143 = 7 \cdot 11^{2} \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 61347 = 3 \cdot 11^{2} \cdot 13^{2} \) |
| $11$ | additive | $62$ | \( 3549 = 3 \cdot 7 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2541 = 3 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 429429.dv consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 273.b1, its twist by $-143$.
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.