Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| 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 $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.81 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&5\\0&11\end{bmatrix}$, $\begin{bmatrix}1&9\\8&9\end{bmatrix}$, $\begin{bmatrix}5&3\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.1.q.2 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $4$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 44x - 112 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{48x^{2}y^{14}-368864x^{2}y^{12}z^{2}+1564511232x^{2}y^{10}z^{4}-118879404186624x^{2}y^{8}z^{6}-35827354485915648x^{2}y^{6}z^{8}+5486059308791169024x^{2}y^{4}z^{10}-236513474548762411008x^{2}y^{2}z^{12}+3279970130870308700160x^{2}z^{14}-1616xy^{14}z+74037504xy^{12}z^{3}-391934689536xy^{10}z^{5}-1242500991135744xy^{8}z^{7}-230282299356348416xy^{6}z^{9}+40179742035806257152xy^{4}z^{11}-1786078758714467155968xy^{2}z^{13}+25114253234762353213440xz^{15}+y^{16}-115584y^{14}z^{2}+1541236992y^{12}z^{4}-9120470863872y^{10}z^{6}-6474529098907648y^{8}z^{8}+136296699068940288y^{6}z^{10}+52469341391619096576y^{4}z^{12}-3078722489027691282432y^{2}z^{14}+47977490845124490428416z^{16}}{zy^{2}(1340x^{2}y^{10}z+6493696x^{2}y^{8}z^{3}+8644488448x^{2}y^{6}z^{5}+4671313813504x^{2}y^{4}z^{7}+1101563548418048x^{2}y^{2}z^{9}+94290337626849280x^{2}z^{11}+xy^{12}+28336xy^{10}z^{2}+87808768xy^{8}z^{4}+92761120768xy^{6}z^{6}+43175964020736xy^{4}z^{8}+9149470046355456xy^{2}z^{10}+721967372344229888xz^{12}+48y^{12}z+443744y^{10}z^{3}+822085632y^{8}z^{5}+582369900544y^{6}z^{7}+187200444301312y^{4}z^{9}+27061695578439680y^{2}z^{11}+1379224087347331072z^{13})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.ba.2.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-16.e.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.0-8.ba.2.6 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.9 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.1-16.a.1.15 | $16$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.192.1-16.a.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.g.2.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.l.2.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.192.1-16.p.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cf.2.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cj.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cv.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.192.1-48.cz.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.288.9-48.ei.2.26 | $48$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.bar.1.25 | $48$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 80.192.1-80.ce.2.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.ci.2.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cu.2.5 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.192.1-80.cy.2.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 80.480.17-80.bw.1.10 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 112.192.1-112.ce.2.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.ci.2.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cu.2.4 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.192.1-112.cy.2.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ce.2.6 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.ci.2.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cu.2.4 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.192.1-176.cy.2.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ce.2.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.ci.2.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cu.2.5 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.192.1-208.cy.2.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.iv.2.14 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.jd.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kb.2.8 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.192.1-240.kj.2.4 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ce.2.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.ci.2.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cu.2.5 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.192.1-272.cy.2.7 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ce.2.6 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.ci.2.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cu.2.4 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.192.1-304.cy.2.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |