Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-1055x+12072\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-1055xz^2+12072z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-85482x+8544069\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(49, 275)$ | $0$ | $4$ |
Integral points
\( \left(24, 0\right) \), \((49,\pm 275)\)
Invariants
| Conductor: | $N$ | = | \( 9240 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $15881250000$ | = | $2^{4} \cdot 3 \cdot 5^{8} \cdot 7 \cdot 11^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{8124052043776}{992578125} \) | = | $2^{11} \cdot 3^{-1} \cdot 5^{-8} \cdot 7^{-1} \cdot 11^{-2} \cdot 1583^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.68764132910482666790845033436$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.45659226891817823143603962721$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9204368882924238$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.5590166766124143$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $1.1974765245741271667825908268$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot1\cdot2^{3}\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.3949530491482543335651816536 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.394953049 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.197477 \cdot 1.000000 \cdot 32}{4^2} \\ & \approx 2.394953049\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8192 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.24.0.45 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18480 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 18465 & 16 \\ 18464 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 13860 & 13861 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 18382 & 18467 \end{array}\right),\left(\begin{array}{rr} 6176 & 5 \\ 18435 & 18466 \end{array}\right),\left(\begin{array}{rr} 3697 & 16 \\ 11096 & 129 \end{array}\right),\left(\begin{array}{rr} 5053 & 16 \\ 1424 & 18165 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 18476 & 18477 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 10576 & 5 \\ 2595 & 18466 \end{array}\right),\left(\begin{array}{rr} 11565 & 16 \\ 16394 & 4859 \end{array}\right)$.
The torsion field $K:=\Q(E[18480])$ is a degree-$78479622144000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18480\Z)$.
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 | $2$ | \( 21 = 3 \cdot 7 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 9240u
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{231}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.5143354480889856.70 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.351298031616.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | add | nonsplit | split | nonsplit | split |
| $\lambda$-invariant(s) | - | 0 | 1 | 0 | 3 |
| $\mu$-invariant(s) | - | 0 | 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 $0$.