Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-x^2-8705x+158797\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-x^2z-8705xz^2+158797z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-139275x+10023750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-101, 50)$ | $0$ | $2$ |
| $(19, -10)$ | $0$ | $2$ |
Integral points
\( \left(-101, 50\right) \), \( \left(19, -10\right) \)
Invariants
| Conductor: | $N$ | = | \( 53550 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $31615317562500$ | = | $2^{2} \cdot 3^{6} \cdot 5^{6} \cdot 7^{4} \cdot 17^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{6403769793}{2775556} \) | = | $2^{-2} \cdot 3^{3} \cdot 7^{-4} \cdot 17^{-2} \cdot 619^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2865806219277094543124691391$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.067444478623395578685533145974$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.1339498614370396$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.566047999291565$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.59354417460117759807449950747$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $4.7483533968094207845959960598 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 4.748353397 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.593544 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 4.748353397\end{aligned}$$
Modular invariants
Modular form 53550.2.a.de
For more coefficients, see the Downloads section to the right.
| Modular degree: | 131072 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $2$ | $I_{2}$ | 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 | 8.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 511 & 1500 \\ 750 & 961 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 751 & 1770 \\ 270 & 271 \end{array}\right),\left(\begin{array}{rr} 241 & 1500 \\ 210 & 961 \end{array}\right),\left(\begin{array}{rr} 1223 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 2037 & 4 \\ 2036 & 5 \end{array}\right),\left(\begin{array}{rr} 679 & 0 \\ 0 & 2039 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$57755566080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \) |
| $5$ | additive | $14$ | \( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 53550dp
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 238c2, its twist by $-15$.
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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{2}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.76785869193364478361600000000.14 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | add | add | nonsplit | split |
| $\lambda$-invariant(s) | 8 | - | - | 0 | 1 |
| $\mu$-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 $0$.