Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.96.1.1843 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&26\\16&37\end{bmatrix}$, $\begin{bmatrix}1&47\\44&5\end{bmatrix}$, $\begin{bmatrix}17&35\\16&23\end{bmatrix}$, $\begin{bmatrix}27&31\\28&7\end{bmatrix}$, $\begin{bmatrix}43&19\\40&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.v.2 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $12288$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x + 14 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(2:0:1)$, $(0:1:0)$, $(1:2:1)$, $(1:-2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{744x^{2}y^{14}-861743506x^{2}y^{12}z^{2}+3946501303224x^{2}y^{10}z^{4}-2404052725370889x^{2}y^{8}z^{6}+450301688638460016x^{2}y^{6}z^{8}-34266169928476438809x^{2}y^{4}z^{10}+1121366330130233819844x^{2}y^{2}z^{12}-13119880523481218023425x^{2}z^{14}-196876xy^{14}z+20032110312xy^{12}z^{3}-40645158196479xy^{10}z^{5}+16765017079652202xy^{8}z^{7}-2481637326273847576xy^{6}z^{9}+160992383698961342616xy^{4}z^{11}-4690996378190219311737xy^{2}z^{13}+50228506469524739981310xz^{15}-y^{16}+21481872y^{14}z^{2}-317842308708y^{12}z^{4}+319718459371944y^{10}z^{6}-82473813056611532y^{8}z^{8}+8224738401779468256y^{6}z^{10}-365602252934307716394y^{4}z^{12}+7147543060259635457184y^{2}z^{14}-47977490845124607868921z^{16}}{y^{2}(x^{2}y^{12}-4x^{2}y^{10}z^{2}-14x^{2}y^{8}z^{4}+16x^{2}y^{6}z^{6}+171x^{2}y^{4}z^{8}+28x^{2}y^{2}z^{10}+x^{2}z^{12}+14xy^{10}z^{3}-100xy^{8}z^{5}+106xy^{6}z^{7}+364xy^{4}z^{9}+57xy^{2}z^{11}+2xz^{13}-16y^{12}z^{2}+112y^{10}z^{4}-177y^{8}z^{6}+200y^{6}z^{8}-1108y^{4}z^{10}-192y^{2}z^{12}-7z^{14})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-8.bb.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-8.bb.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.2.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.0-16.e.2.10 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.48.1-16.b.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.48.1-16.b.1.15 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.192.1-16.f.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-16.i.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-16.m.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-16.s.2.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-16.dz.2.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-16.ea.1.3 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-16.eb.1.3 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-16.ec.2.7 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 96.192.3-32.z.2.13 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-32.ba.2.9 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-32.bb.2.11 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-32.bc.1.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.5-32.s.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-32.w.2.14 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-32.ba.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-32.be.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 48.192.1-48.co.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cs.2.10 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.de.1.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.di.1.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.3-48.kt.2.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.ku.1.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.kv.1.6 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.192.3-48.kw.2.10 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.288.9-48.ez.2.37 | $48$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bba.2.39 | $48$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 240.192.1-80.cn.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-80.cr.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-80.dd.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-80.dh.1.5 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-80.nl.2.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.nm.2.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.nn.2.12 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-80.no.2.14 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.480.17-80.cf.2.22 | $240$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 96.192.3-96.bh.2.31 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bi.2.31 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bj.2.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.3-96.bk.1.15 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.192.5-96.bu.2.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.by.2.27 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cs.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.192.5-96.cw.2.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.1-240.jm.1.23 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ju.2.19 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.ks.1.6 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.la.1.21 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.3-240.bld.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.ble.2.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.blf.1.10 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.192.3-240.blg.2.6 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |