Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-20701287x+36244390329\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-20701287xz^2+36244390329z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-26828868627x+1691420708215854\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1953, 56196)$ | $1.0485487574487070890542539275$ | $\infty$ |
| $(2558, 4771)$ | $0$ | $4$ |
Integral points
\( \left(1953, 56196\right) \), \( \left(1953, -58149\right) \), \( \left(2558, 4771\right) \), \( \left(2558, -7329\right) \), \( \left(2628, -1179\right) \), \( \left(2628, -1449\right) \), \( \left(2933, 26271\right) \), \( \left(2933, -29204\right) \), \( \left(6408, 405171\right) \), \( \left(6408, -411579\right) \)
Invariants
| Conductor: | $N$ | = | \( 3630 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $22197105717187500$ | = | $2^{2} \cdot 3^{6} \cdot 5^{8} \cdot 11^{7} $ |
|
| j-invariant: | $j$ | = | \( \frac{553808571467029327441}{12529687500} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-8} \cdot 11^{-1} \cdot 23^{3} \cdot 357047^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6607072599721155228295282618$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4617596235729302507985564728$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.047398607062714$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.582141507390664$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0485487574487070890542539275$ |
|
| Real period: | $\Omega$ | ≈ | $0.27612060320318187220343547362$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2\cdot2^{3}\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.3162073231574707410803964607 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.316207323 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.276121 \cdot 1.048549 \cdot 128}{4^2} \\ & \approx 2.316207323\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 184320 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $4$ | $I_{1}^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.24.0.46 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1057 & 16 \\ 536 & 129 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 404 & 345 \end{array}\right),\left(\begin{array}{rr} 952 & 2639 \\ 161 & 2630 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 2542 & 2627 \end{array}\right),\left(\begin{array}{rr} 2318 & 661 \\ 409 & 1330 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 893 & 16 \\ 1504 & 2325 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 2636 & 2637 \end{array}\right),\left(\begin{array}{rr} 2625 & 16 \\ 2624 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[2640])$ is a degree-$38928384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2640\Z)$.
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 | $4$ | \( 121 = 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 3630d
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330c5, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.85184.1 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.587761422336.37 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.116101021696.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | nonsplit | nonsplit | split | ss | add | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 3 | 2 | 1,3 | - | 1 | 1 | 3 | 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 | 0 | 0 |
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.