Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-1250235065x+17014724166663\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-1250235065xz^2+17014724166663z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-101269040292x+12404037724618176\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(19921, 121500)$ | $0.68362820559912959013205936738$ | $\infty$ |
| $(20407, 0)$ | $0$ | $2$ |
| $(20421, 0)$ | $0$ | $2$ |
Integral points
\( \left(-40829, 0\right) \), \((-21929,\pm 5821200)\), \((14641,\pm 1359660)\), \((18706,\pm 416745)\), \((19921,\pm 121500)\), \( \left(20407, 0\right) \), \( \left(20421, 0\right) \), \((20911,\pm 123480)\), \((22171,\pm 441000)\), \((157621,\pm 61122600)\)
Invariants
| Conductor: | $N$ | = | \( 47040 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7^{2}$ |
|
| Discriminant: | $\Delta$ | = | $44116583158670400000000$ | = | $2^{12} \cdot 3^{14} \cdot 5^{8} \cdot 7^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{448487713888272974160064}{91549016015625} \) | = | $2^{6} \cdot 3^{-14} \cdot 5^{-8} \cdot 7^{-2} \cdot 19136251^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7329426324155456697255461393$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.0668403773279437077556376461$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0964009209698897$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.920262159537654$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.68362820559912959013205936738$ |
|
| Real period: | $\Omega$ | ≈ | $0.090160347545950413662599100443$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1792 $ = $ 2^{2}\cdot( 2 \cdot 7 )\cdot2^{3}\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.9032495402115804588444121983 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.903249540 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.090160 \cdot 0.683628 \cdot 1792}{4^2} \\ & \approx 6.903249540\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20643840 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | -1 | 6 | 12 | 0 |
| $3$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 45 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 95 & 166 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 110 & 167 \end{array}\right),\left(\begin{array}{rr} 81 & 166 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 165 & 4 \\ 164 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$3096576$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 49 = 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 15680 = 2^{6} \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 320 = 2^{6} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 47040hi
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3360q1, its twist by $56$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.162447943996702457856.1 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | split | split | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | 4 | - | 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 |
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.