Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-5001x+134440\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-5001xz^2+134440z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-405108x+99222057\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(29, 121)$ | $0.48550621440824240087915546991$ | $\infty$ |
| $(40, 0)$ | $0$ | $2$ |
Integral points
\((-81,\pm 121)\), \((29,\pm 121)\), \( \left(40, 0\right) \), \((41,\pm 1)\), \((51,\pm 121)\), \((3761,\pm 230641)\)
Invariants
| Conductor: | $N$ | = | \( 2420 \) | = | $2^{2} \cdot 5 \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $3543122000$ | = | $2^{4} \cdot 5^{3} \cdot 11^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{488095744}{125} \) | = | $2^{14} \cdot 5^{-3} \cdot 31^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.81830368521772014655456832687$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.61169301136811356194881416927$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.073761143790618$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.770053731855172$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.48550621440824240087915546991$ |
|
| Real period: | $\Omega$ | ≈ | $1.3713732139096676920398953522$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 3\cdot1\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $1.9974306528784427775247617405 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 1.997430653 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.371373 \cdot 0.485506 \cdot 12}{2^2} \\ & \approx 1.997430653\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2160 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_0^{*}$ | 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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 7 & 24 \\ 492 & 367 \end{array}\right),\left(\begin{array}{rr} 705 & 946 \\ 374 & 155 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 1220 & 1301 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1297 & 24 \\ 1296 & 25 \end{array}\right),\left(\begin{array}{rr} 661 & 264 \\ 792 & 529 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 232 & 363 \\ 957 & 34 \end{array}\right),\left(\begin{array}{rr} 881 & 264 \\ 880 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 0 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1216512000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $2$ | \( 605 = 5 \cdot 11^{2} \) |
| $3$ | good | $2$ | \( 484 = 2^{2} \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 484 = 2^{2} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 20 = 2^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 2420.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 20.a1, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/6\Z\) | 2.2.33.1-400.1-a4 |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.38720.4 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.15524784.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.937024000000.10 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.37480960000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.121438310400.3 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.2169170264219904.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.17980759982220503815017000000000000.3 | \(\Z/18\Z\) | 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 | ord | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 1 | 3 | - | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.