Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3-x^2-13567x-525159\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3-x^2z-13567xz^2-525159z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-217075x-33827250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-72, 309)$ | $1.9027701223287647018003926633$ | $\infty$ |
| $(-86, 43)$ | $0$ | $2$ |
| $(-46, 23)$ | $0$ | $2$ |
Integral points
\( \left(-86, 43\right) \), \( \left(-72, 309\right) \), \( \left(-72, -237\right) \), \( \left(-46, 23\right) \), \( \left(229, 2773\right) \), \( \left(229, -3002\right) \), \( \left(539, 11918\right) \), \( \left(539, -12457\right) \)
Invariants
| Conductor: | $N$ | = | \( 50050 \) | = | $2 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $39140664062500$ | = | $2^{2} \cdot 5^{10} \cdot 7^{2} \cdot 11^{2} \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{17675559395649}{2505002500} \) | = | $2^{-2} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{-2} \cdot 11^{-2} \cdot 13^{-2} \cdot 19^{3} \cdot 457^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3340750049178193893676728320$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52935604870076920206729316539$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.904026026242857$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7113627365505804$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9027701223287647018003926633$ |
|
| Real period: | $\Omega$ | ≈ | $0.44653415639584617657530528185$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.3986074055571839138663897676 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.398607406 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.446534 \cdot 1.902770 \cdot 64}{4^2} \\ & \approx 3.398607406\end{aligned}$$
Modular invariants
Modular form 50050.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 172032 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 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 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 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 | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 40040 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 11441 & 8010 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36401 & 16020 \\ 730 & 32041 \end{array}\right),\left(\begin{array}{rr} 36961 & 16020 \\ 1850 & 32041 \end{array}\right),\left(\begin{array}{rr} 28031 & 8010 \\ 32030 & 32031 \end{array}\right),\left(\begin{array}{rr} 40037 & 4 \\ 40036 & 5 \end{array}\right),\left(\begin{array}{rr} 18021 & 8010 \\ 8000 & 16011 \end{array}\right),\left(\begin{array}{rr} 24023 & 0 \\ 0 & 40039 \end{array}\right)$.
The torsion field $K:=\Q(E[40040])$ is a degree-$10712468422656000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40040\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $5$ | additive | $18$ | \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 7150 = 2 \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 50050p
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 10010m2, its twist by $5$.
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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{10}, \sqrt{-286})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{35}, \sqrt{715})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | ss | add | split | nonsplit | split | ord | ord | ss | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 1,3 | - | 2 | 3 | 2 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 3,1 |
| $\mu$-invariant(s) | 0 | 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.