Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-22032075x+39804412250\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-22032075xz^2+39804412250z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-22032075x+39804412250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2714, 362)$ | $4.0259923850876217257166681560$ | $\infty$ |
| $(2695, 1350)$ | $0$ | $4$ |
Integral points
\((-2705,\pm 282150)\), \((2695,\pm 1350)\), \( \left(2710, 0\right) \), \((2714,\pm 362)\)
Invariants
| Conductor: | $N$ | = | \( 61200 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 17$ |
|
| Discriminant: | $\Delta$ | = | $53537760000000$ | = | $2^{11} \cdot 3^{9} \cdot 5^{7} \cdot 17 $ |
|
| j-invariant: | $j$ | = | \( \frac{50700519510140162}{2295} \) | = | $2 \cdot 3^{-3} \cdot 5^{-1} \cdot 13^{3} \cdot 17^{-1} \cdot 59^{3} \cdot 383^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5633059425862974606831132275$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.57389592652190922738598149775$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0016136042307515$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.655796077231814$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.0259923850876217257166681560$ |
|
| Real period: | $\Omega$ | ≈ | $0.34077288762702002113726888157$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $5.4877962025228097353300611390 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 5.487796203 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.340773 \cdot 4.025992 \cdot 64}{4^2} \\ & \approx 5.487796203\end{aligned}$$
Modular invariants
Modular form 61200.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.12.0.7 | $12$ |
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
$\left(\begin{array}{rr} 7 & 6 \\ 2034 & 2035 \end{array}\right),\left(\begin{array}{rr} 1272 & 247 \\ 1247 & 1200 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 812 & 2039 \\ 1609 & 2034 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1928 & 3 \\ 1205 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 256 & 1283 \\ 769 & 798 \end{array}\right),\left(\begin{array}{rr} 2033 & 8 \\ 2032 & 9 \end{array}\right),\left(\begin{array}{rr} 676 & 2039 \\ 1337 & 2034 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \) |
| $3$ | additive | $6$ | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 61200.u
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040.d1, its twist by $60$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{510}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.459000.4 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3896755776000000.20 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | add | ord | ord | ord | split | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 1 | 1 | 1 | 2 | 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.