Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-1338892969x+18856640041292\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-1338892969xz^2+18856640041292z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-1735205287203x+879780603382392798\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(20158, 231662)$ | $1.9638461191014653676077388087$ | $\infty$ |
| $(84519/4, -84523/8)$ | $0$ | $2$ |
Integral points
\( \left(20158, 231662\right) \), \( \left(20158, -251821\right) \), \( \left(681286, 561198419\right) \), \( \left(681286, -561879706\right) \)
Invariants
| Conductor: | $N$ | = | \( 53130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$ |
|
| Discriminant: | $\Delta$ | = | $17149152760523437500000$ | = | $2^{5} \cdot 3^{3} \cdot 5^{12} \cdot 7^{4} \cdot 11^{2} \cdot 23^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{265436898662503851515370589836169}{17149152760523437500000} \) | = | $2^{-5} \cdot 3^{-3} \cdot 5^{-12} \cdot 7^{-4} \cdot 11^{-2} \cdot 23^{-4} \cdot 64266862489^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7249331186652904158859260806$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.7249331186652904158859260806$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0213780810476365$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.8617205444758875$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9638461191014653676077388087$ |
|
| Real period: | $\Omega$ | ≈ | $0.093198967778781216176544035774$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 1\cdot3\cdot2\cdot2\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.3926823482389282271538047373 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.392682348 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.093199 \cdot 1.963846 \cdot 96}{2^2} \\ & \approx 4.392682348\end{aligned}$$
Modular invariants
Modular form 53130.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 23592960 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $23$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 8.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1844 & 1 \\ 943 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1657 & 8 \\ 1108 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2416 & 1043 \\ 1729 & 1758 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2754 & 2755 \end{array}\right),\left(\begin{array}{rr} 1728 & 353 \\ 1753 & 1800 \end{array}\right),\left(\begin{array}{rr} 1201 & 8 \\ 2044 & 33 \end{array}\right),\left(\begin{array}{rr} 2753 & 8 \\ 2752 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[2760])$ is a degree-$196977623040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2760\Z)$.
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 | $4$ | \( 3 \) |
| $3$ | split multiplicative | $4$ | \( 3542 = 2 \cdot 7 \cdot 11 \cdot 23 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5313 = 3 \cdot 7 \cdot 11 \cdot 23 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 7590 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 23 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| $23$ | split multiplicative | $24$ | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 53130.s
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.44767018745856.110 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.522250467840000.11 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 |
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 | split | nonsplit | nonsplit | nonsplit | ord | ord | ord | split | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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.