Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+486933x-116586722\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+486933xz^2-116586722z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+631065789x-5441363287362\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7094, 596748)$ | $5.1057690941302334133993756430$ | $\infty$ |
Integral points
\( \left(7094, 596748\right) \), \( \left(7094, -603843\right) \)
Invariants
| Conductor: | $N$ | = | \( 422142 \) | = | $2 \cdot 3 \cdot 7 \cdot 19 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-13265087146454591784$ | = | $-1 \cdot 2^{3} \cdot 3^{2} \cdot 7^{3} \cdot 19^{3} \cdot 23^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{163045918103}{169389864} \) | = | $2^{-3} \cdot 3^{-2} \cdot 7^{-3} \cdot 17^{3} \cdot 19^{-3} \cdot 23 \cdot 113^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3560747031741180270070907773$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.26574522588801823313592222276$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8750555597277171$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.929658777086063$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1057690941302334133993756430$ |
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| Real period: | $\Omega$ | ≈ | $0.12142422399573551974399835829$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot2\cdot3\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.159352902811115722010835186 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.159352903 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.121424 \cdot 5.105769 \cdot 18}{1^2} \\ & \approx 11.159352903\end{aligned}$$
Modular invariants
Modular form 422142.2.a.cc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11446272 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $23$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
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 |
|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 913 & 6 \\ 2739 & 19 \end{array}\right),\left(\begin{array}{rr} 799 & 6 \\ 2397 & 19 \end{array}\right),\left(\begin{array}{rr} 3189 & 3190 \\ 3182 & 3185 \end{array}\right),\left(\begin{array}{rr} 1597 & 6 \\ 1599 & 19 \end{array}\right),\left(\begin{array}{rr} 3187 & 6 \\ 3186 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1466 & 1731 \\ 1207 & 1603 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1009 & 6 \\ 3027 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$1143751311360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $4$ | \( 70357 = 7 \cdot 19 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 529 = 23^{2} \) |
| $7$ | split multiplicative | $8$ | \( 60306 = 2 \cdot 3 \cdot 19 \cdot 23^{2} \) |
| $19$ | split multiplicative | $20$ | \( 22218 = 2 \cdot 3 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $200$ | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 422142cc
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 422142v2, its twist by $-23$.
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.