Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-27824x+2125236\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-27824xz^2+2125236z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2253771x+1556058330\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(46, 972)$ | $0.65232743212421312136853448363$ | $\infty$ |
| $(78, 660)$ | $1.9595677597193463983728970732$ | $\infty$ |
| $(100, 594)$ | $0$ | $4$ |
Integral points
\( \left(-197, 0\right) \), \((-170,\pm 1404)\), \((-164,\pm 1518)\), \((-98,\pm 1980)\), \((-65,\pm 1914)\), \((28,\pm 1170)\), \((46,\pm 972)\), \((78,\pm 660)\), \((100,\pm 594)\), \((127,\pm 810)\), \((166,\pm 1452)\), \((244,\pm 3150)\), \((310,\pm 4836)\), \((331,\pm 5412)\), \((694,\pm 17820)\), \((2476,\pm 122958)\), \((3070,\pm 169884)\), \((14086,\pm 1671732)\), \((237502,\pm 115744860)\)
Invariants
| Conductor: | $N$ | = | \( 19536 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 37$ |
|
| Discriminant: | $\Delta$ | = | $-589599998355456$ | = | $-1 \cdot 2^{11} \cdot 3^{12} \cdot 11^{4} \cdot 37 $ |
|
| j-invariant: | $j$ | = | \( -\frac{1163236610689634}{287890624197} \) | = | $-1 \cdot 2 \cdot 3^{-12} \cdot 11^{-4} \cdot 13^{3} \cdot 37^{-1} \cdot 6421^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5522832723337098177512537377$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.91689835682042661745212429303$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9461181003915472$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.318884641648152$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2212849715670262831924813343$ |
|
| Real period: | $\Omega$ | ≈ | $0.49158216868838303824579046253$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.2043429789133879627517036854 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.204342979 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.491582 \cdot 1.221285 \cdot 192}{4^2} \\ & \approx 7.204342979\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 98304 |
|
| $ \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 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $11$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $37$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9768 = 2^{3} \cdot 3 \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9761 & 8 \\ 9760 & 9 \end{array}\right),\left(\begin{array}{rr} 6104 & 1213 \\ 6101 & 6072 \end{array}\right),\left(\begin{array}{rr} 3257 & 8 \\ 3260 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9762 & 9763 \end{array}\right),\left(\begin{array}{rr} 6076 & 1 \\ 3191 & 6 \end{array}\right),\left(\begin{array}{rr} 1224 & 6113 \\ 8539 & 8526 \end{array}\right),\left(\begin{array}{rr} 6217 & 8 \\ 5332 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[9768])$ is a degree-$36944982835200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9768\Z)$.
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 | $4$ | \( 37 \) |
| $3$ | split multiplicative | $4$ | \( 6512 = 2^{4} \cdot 11 \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 1776 = 2^{4} \cdot 3 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 19536p
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 9768n4, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-74}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.322344.3 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.43045746160697344.3 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.9103797810302976.3 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | ord | ord | split | ord | ord | ord | ss | ord | ss | split | ord | ord | ord |
| $\lambda$-invariant(s) | - | 5 | 2 | 2 | 3 | 2 | 2 | 2 | 2,2 | 2 | 2,2 | 3 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.