Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-17997x-863343\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-17997xz^2-863343z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-23324139x-40210158618\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-94, 47)$ | $0$ | $2$ |
Integral points
\( \left(-94, 47\right) \)
Invariants
| Conductor: | $N$ | = | \( 53958 \) | = | $2 \cdot 3 \cdot 17 \cdot 23^{2}$ |
|
| Discriminant: | $\Delta$ | = | $52184427303168$ | = | $2^{8} \cdot 3^{4} \cdot 17 \cdot 23^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{4354703137}{352512} \) | = | $2^{-8} \cdot 3^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 71^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3751046227661291355153552904$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.19264248519844570988802112550$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.051915192059638$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.763549323632871$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.41431057520543001811145709013$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{3}\cdot2^{2}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $6.6289692032868802897833134420 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 6.628969203 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.414311 \cdot 1.000000 \cdot 64}{2^2} \\ & \approx 6.628969203\end{aligned}$$
Modular invariants
Modular form 53958.2.a.bn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 202752 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 16.48.0.101 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6256 = 2^{4} \cdot 17 \cdot 23 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1104 \\ 4692 & 4693 \end{array}\right),\left(\begin{array}{rr} 24 & 1633 \\ 4623 & 2186 \end{array}\right),\left(\begin{array}{rr} 271 & 0 \\ 0 & 6255 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 6158 & 6243 \end{array}\right),\left(\begin{array}{rr} 1473 & 1104 \\ 506 & 5935 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 6252 & 6253 \end{array}\right),\left(\begin{array}{rr} 6241 & 16 \\ 6240 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[6256])$ is a degree-$2678895673344$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6256\Z)$.
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 | $4$ | \( 8993 = 17 \cdot 23^{2} \) |
| $3$ | split multiplicative | $4$ | \( 17986 = 2 \cdot 17 \cdot 23^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 3174 = 2 \cdot 3 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 53958bh
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102b1, its twist by $-23$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-391}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.4.1729198450311424.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1729198450311424.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.26832068481024.17 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | 17 | 23 |
|---|---|---|---|---|
| Reduction type | split | split | nonsplit | add |
| $\lambda$-invariant(s) | 7 | 1 | 0 | - |
| $\mu$-invariant(s) | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.