Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-50420535x-85142565486\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-50420535xz^2-85142565486z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-65345014035x-3971431360107666\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(809898190035/94439524, 304931592286096479/917763294232)$ | $22.187874875963343691166686002$ | $\infty$ |
| $(-7221/4, 7221/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 467313 \) | = | $3 \cdot 7^{2} \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $5073426046805839227192303$ | = | $3^{16} \cdot 7^{9} \cdot 11^{2} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{14553591673375}{5208653241} \) | = | $3^{-16} \cdot 5^{3} \cdot 11^{-2} \cdot 19^{3} \cdot 257^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4413674442413983983036966191$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.56532816042180537934991475258$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0123792461665873$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.965342710851245$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $22.187874875963343691166686002$ |
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| Real period: | $\Omega$ | ≈ | $0.058337585639795327925597517955$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1775482029662988068269764984 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.177548203 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.058338 \cdot 22.187875 \cdot 16}{2^2} \\ & \approx 5.177548203\end{aligned}$$
Modular invariants
Modular form 467313.2.a.bp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 88309760 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $7$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 8.12.0.27 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 616 = 2^{3} \cdot 7 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 608 & 595 \end{array}\right),\left(\begin{array}{rr} 609 & 8 \\ 608 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 162 \\ 154 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 360 & 3 \\ 221 & 600 \end{array}\right),\left(\begin{array}{rr} 466 & 465 \\ 21 & 478 \end{array}\right),\left(\begin{array}{rr} 285 & 2 \\ 22 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[616])$ is a degree-$851558400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/616\Z)$.
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 | $2$ | \( 2023 = 7 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 155771 = 7^{2} \cdot 11 \cdot 17^{2} \) |
| $7$ | additive | $20$ | \( 9537 = 3 \cdot 11 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 42483 = 3 \cdot 7^{2} \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 467313bp
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1617d2, its twist by $-119$.
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.