Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-16391998297x-807792164146877\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-16391998297xz^2-807792164146877z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-21244029793587x-37688032549989792882\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-977582661/13225, 56210902508/1520875)$ | $10.110471679195337369878403789$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 459186 \) | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $6661478332075382634$ | = | $2 \cdot 3 \cdot 7 \cdot 13 \cdot 29^{11} $ |
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| j-invariant: | $j$ | = | \( \frac{818901045522640857815176321}{11199087354} \) | = | $2^{-1} \cdot 3^{-1} \cdot 7^{-1} \cdot 13^{-1} \cdot 29^{-5} \cdot 389^{3} \cdot 2405069^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0992682500533595353237217706$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4156203350601225217320857544$ |
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| $abc$ quality: | $Q$ | ≈ | $1.010738907345902$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.303018549049233$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.110471679195337369878403789$ |
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| Real period: | $\Omega$ | ≈ | $0.013342634199444277789522609865$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7450162599672261600673078449 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $25$ = $5^2$ (rounded) |
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BSD formula
$$\begin{aligned} 6.745016260 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.013343 \cdot 10.110472 \cdot 2}{1^2} \\ & \approx 6.745016260\end{aligned}$$
Modular invariants
Modular form 459186.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 336000000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $2$ | $I_{5}^{*}$ | additive | 1 | 2 | 11 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 316680 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 29 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 6 & 13 \\ 316625 & 316561 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 316671 & 10 \\ 316670 & 11 \end{array}\right),\left(\begin{array}{rr} 237511 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 146161 & 10 \\ 97445 & 51 \end{array}\right),\left(\begin{array}{rr} 226201 & 10 \\ 180965 & 51 \end{array}\right),\left(\begin{array}{rr} 79177 & 158350 \\ 316640 & 269121 \end{array}\right),\left(\begin{array}{rr} 98279 & 316670 \\ 174715 & 316629 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 211121 & 10 \\ 105565 & 51 \end{array}\right),\left(\begin{array}{rr} 158341 & 10 \\ 158345 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[316680])$ is a degree-$26570038042637107200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/316680\Z)$.
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 | $4$ | \( 229593 = 3 \cdot 7 \cdot 13 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 153062 = 2 \cdot 7 \cdot 13 \cdot 29^{2} \) |
| $7$ | split multiplicative | $8$ | \( 65598 = 2 \cdot 3 \cdot 13 \cdot 29^{2} \) |
| $13$ | split multiplicative | $14$ | \( 35322 = 2 \cdot 3 \cdot 7 \cdot 29^{2} \) |
| $29$ | additive | $450$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 459186q
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 15834t2, its twist by $29$.
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.