Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-6541x-206341\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-6541xz^2-206341z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-8477163x-9499879962\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-47, 24)$ | $1.3695910777401621016314385092$ | $\infty$ |
| $(-189/4, 185/8)$ | $0$ | $2$ |
Integral points
\( \left(-47, 24\right) \), \( \left(-47, 22\right) \), \( \left(109, 570\right) \), \( \left(109, -680\right) \), \( \left(333, 5722\right) \), \( \left(333, -6056\right) \)
Invariants
| Conductor: | $N$ | = | \( 510 \) | = | $2 \cdot 3 \cdot 5 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $478125000$ | = | $2^{3} \cdot 3^{2} \cdot 5^{8} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{30949975477232209}{478125000} \) | = | $2^{-3} \cdot 3^{-2} \cdot 5^{-8} \cdot 17^{-1} \cdot 313969^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.80053291371084970050409631005$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80053291371084970050409631005$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0024850789671809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.090575400344785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3695910777401621016314385092$ |
|
| Real period: | $\Omega$ | ≈ | $0.53087047736269429338675037738$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 3\cdot2\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.1812264076948204153605107107 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.181226408 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.530870 \cdot 1.369591 \cdot 12}{2^2} \\ & \approx 2.181226408\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 768 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 8.24.0.102 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 136 = 2^{3} \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 120 & 59 \\ 89 & 118 \end{array}\right),\left(\begin{array}{rr} 123 & 120 \\ 74 & 125 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 129 & 8 \\ 128 & 9 \end{array}\right),\left(\begin{array}{rr} 12 & 1 \\ 103 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 130 & 131 \end{array}\right)$.
The torsion field $K:=\Q(E[136])$ is a degree-$2506752$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/136\Z)$.
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 | $4$ | \( 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 85 = 5 \cdot 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 510d
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{34}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.34816.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.8200464478765056.39 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.8008266092544.38 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1401249857536.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.9247184116875.4 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | nonsplit | nonsplit | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.