Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2+71592x-11228688\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z+71592xz^2-11228688z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+5798925x-8168316750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(786, 23022)$ | $5.0642198252960747849987552509$ | $\infty$ |
Integral points
\((786,\pm 23022)\)
Invariants
| Conductor: | $N$ | = | \( 145200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-77720518656000000$ | = | $-1 \cdot 2^{22} \cdot 3^{4} \cdot 5^{6} \cdot 11^{4} $ |
|
| j-invariant: | $j$ | = | \( \frac{43307231}{82944} \) | = | $2^{-10} \cdot 3^{-4} \cdot 11^{2} \cdot 71^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9255860808888547583637638580$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37157848015426425304116245606$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0632893712610214$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.869036561735465$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.0642198252960747849987552509$ |
|
| Real period: | $\Omega$ | ≈ | $0.17953407280875744236209799590$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.6368000433370335409329409217 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.636800043 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.179534 \cdot 5.064220 \cdot 4}{1^2} \\ & \approx 3.636800043\end{aligned}$$
Modular invariants
Modular form 145200.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1612800 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{14}^{*}$ | additive | -1 | 4 | 22 | 10 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 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 |
|---|---|---|
| $2$ | 2G | 4.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 20.16.0-4.b.1.1, level \( 20 = 2^{2} \cdot 5 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 17 & 4 \\ 16 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 19 \end{array}\right),\left(\begin{array}{rr} 9 & 5 \\ 0 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[20])$ is a degree-$2880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.
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 | $2$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $14$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $52$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 145200.g consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 726.f1, its twist by $-20$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.484.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.937024.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.29282000.2 | \(\Z/4\Z\) | not in database |
| $6$ | 6.0.117128000.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | 12.0.13718968384000000.2 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | nonsplit | add | ord | add | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 3,1 | 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.