Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-17208x+867901\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-17208xz^2+867901z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-275331x+55270334\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(20, 719\right) \) | $1.3192544115670790874427516950$ | $\infty$ |
| \( \left(\frac{323}{4}, -\frac{323}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([20:719:1]\) | $1.3192544115670790874427516950$ | $\infty$ |
| \([646:-323:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(79, 5832\right) \) | $1.3192544115670790874427516950$ | $\infty$ |
| \( \left(322, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(20, 719\right) \), \( \left(20, -739\right) \), \( \left(44, 419\right) \), \( \left(44, -463\right) \)
\([20:719:1]\), \([20:-739:1]\), \([44:419:1]\), \([44:-463:1]\)
\((79,\pm 5832)\), \((175,\pm 3528)\)
Invariants
| Conductor: | $N$ | = | \( 441 \) | = | $3^{2} \cdot 7^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $3938980639167$ | = | $3^{14} \cdot 7^{7} $ |
|
| j-invariant: | $j$ | = | \( \frac{6570725617}{45927} \) | = | $3^{-8} \cdot 7^{-1} \cdot 1873^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2498930119726786142999487841$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.27236820688903288395035020608$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0015974661127929$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.712551076522215$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3192544115670790874427516950$ |
|
| Real period: | $\Omega$ | ≈ | $0.78753161613464932317444500209$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.0779091176683751994920069133 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.077909118 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.787532 \cdot 1.319254 \cdot 8}{2^2} \\ & \approx 2.077909118\end{aligned}$$
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 not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.48.0.99 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 336 = 2^{4} \cdot 3 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 332 & 333 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 238 & 323 \end{array}\right),\left(\begin{array}{rr} 321 & 16 \\ 320 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 92 & 213 \end{array}\right),\left(\begin{array}{rr} 218 & 263 \\ 43 & 54 \end{array}\right),\left(\begin{array}{rr} 223 & 320 \\ 104 & 207 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 88 & 335 \\ 113 & 326 \end{array}\right)$.
The torsion field $K:=\Q(E[336])$ is a degree-$12386304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/336\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 49 = 7^{2} \) |
| $7$ | additive | $32$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 441c
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21a3, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.28.1-567.1-c5 |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/8\Z\) | 2.2.21.1-21.1-b4 |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.624529833984.34 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.624529833984.36 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.49787136.3 | \(\Z/16\Z\) | not in database |
| $8$ | 8.8.624529833984.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.2.20841167403.2 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.634562281237118976.3 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | ord | add | ord | add | ord | ord | ord | ord | ss | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | ? | - | 3 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | ? | - | 0 | - | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.