Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-578448x+183565278\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-578448xz^2+183565278z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-749667987x+8566670625966\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(234, 7695)$ | $0$ | $3$ |
Integral points
\( \left(234, 7695\right) \), \( \left(234, -7930\right) \)
Invariants
| Conductor: | $N$ | = | \( 910 \) | = | $2 \cdot 5 \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-2177246093750000000$ | = | $-1 \cdot 2^{7} \cdot 5^{18} \cdot 7^{3} \cdot 13 $ |
|
| j-invariant: | $j$ | = | \( -\frac{21405018343206000779641}{2177246093750000000} \) | = | $-1 \cdot 2^{-7} \cdot 5^{-18} \cdot 7^{-3} \cdot 13^{-1} \cdot 53^{3} \cdot 523877^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2585354888037067037533041064$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2585354888037067037533041064$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0145266852097035$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.570301881260937$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.25379443858147822808785741497$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 1\cdot( 2 \cdot 3^{2} )\cdot3\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.5227666314888693685271444898 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.522766631 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.253794 \cdot 1.000000 \cdot 54}{3^2} \\ & \approx 1.522766631\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 21168 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $5$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3277 & 18 \\ 3285 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 5617 & 18 \\ 4689 & 163 \end{array}\right),\left(\begin{array}{rr} 1649 & 3294 \\ 3114 & 4285 \end{array}\right),\left(\begin{array}{rr} 1639 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right),\left(\begin{array}{rr} 13 & 18 \\ 3897 & 511 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$2191186722816$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
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 | $4$ | \( 91 = 7 \cdot 13 \) |
| $3$ | good | $2$ | \( 26 = 2 \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 182 = 2 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 65 = 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 910.e
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.728.1 | \(\Z/6\Z\) | not in database |
| $3$ | 3.3.169.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.12338352.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.73008.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.11019643561472.6 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.1878328153971890270208.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.611880132587388554601258477944832.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.21423624263414745793258586112.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.277231469918931622716761830326272.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | split | split | split |
| $\lambda$-invariant(s) | 2 | 2 | 1 | 3 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.