Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $2 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 14 }{2}$ | ||||||
| Cusps: | $14$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{6}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}\cdot4$ | ||
| 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: | 16K2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.2.320 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&10\\40&41\end{bmatrix}$, $\begin{bmatrix}11&46\\32&39\end{bmatrix}$, $\begin{bmatrix}17&2\\0&41\end{bmatrix}$, $\begin{bmatrix}35&22\\24&7\end{bmatrix}$, $\begin{bmatrix}37&12\\32&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.2.e.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{4}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 576.2.k.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x w^{2} + y z w $ |
| $=$ | $x z w + y z^{2}$ | |
| $=$ | $x y w + y^{2} z$ | |
| $=$ | $x^{2} w + x y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{3} y^{2} - x^{3} z^{2} - 3 x^{2} y^{2} z + x^{2} z^{3} - 9 x y^{2} z^{2} - x z^{4} - 3 y^{2} z^{3} + z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 3x^{5} + 6x^{4} + 6x^{2} - 3x $ |
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.
| Embedded model |
|---|
| $(0:0:1:1)$, $(1:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{90999702524887200x^{2}y^{18}+80590980797194032x^{2}y^{16}w^{2}+14057938041470640x^{2}y^{14}w^{4}-15293131702708608x^{2}y^{12}w^{6}-21147438262900608x^{2}y^{10}w^{8}-17113079911738464x^{2}y^{8}w^{10}-11285287901981856x^{2}y^{6}w^{12}-6624981725309568x^{2}y^{4}w^{14}-3599428681062432x^{2}y^{2}w^{16}-1848744150934992x^{2}w^{18}-128693012525809344xy^{19}-144306095130733536xy^{17}w^{2}-54507535355404224xy^{15}w^{4}+7923071255129856xy^{13}w^{6}+31284795182328000xy^{11}w^{8}+31486902134651712xy^{9}w^{10}+23510363715140544xy^{7}w^{12}+15096247409373696xy^{5}w^{14}+8809736357975040xy^{3}w^{16}+4804937612924832xyw^{18}+37693309989584736y^{20}+28177519245129360y^{18}w^{2}+600571383911856y^{16}w^{4}-7781002733378880y^{14}w^{6}-8088125498807040y^{12}w^{8}-5814328336870752y^{10}w^{10}-3545194884658848y^{8}w^{12}-1958861330995392y^{6}w^{14}-1012077253341792y^{4}w^{16}-497717863002672y^{2}w^{18}-z^{20}-5z^{19}w-737z^{18}w^{2}-3675z^{17}w^{3}-193214z^{16}w^{4}-958432z^{15}w^{5}-20934584z^{14}w^{6}-102544904z^{13}w^{7}-850356378z^{12}w^{8}-3991878498z^{11}w^{9}-21920310066z^{10}w^{10}-95815386918z^{9}w^{11}-427596391572z^{8}w^{12}-1728029469576z^{7}w^{13}-6846981737712z^{6}w^{14}-25766763942448z^{5}w^{15}-94395979121933z^{4}w^{16}-334539291676721z^{3}w^{17}+343140236966315z^{2}w^{18}-310408632432967zw^{19}+431095740373994w^{20}}{w^{2}(629856x^{2}y^{14}w^{2}+1924560x^{2}y^{12}w^{4}-314928x^{2}y^{10}w^{6}-3133728x^{2}y^{8}w^{8}-145152x^{2}y^{6}w^{10}+1383696x^{2}y^{4}w^{12}-2545776x^{2}y^{2}w^{14}-5977344x^{2}w^{16}+1259712xy^{15}w^{2}+4269024xy^{13}w^{4}+513216xy^{11}w^{6}-6578496xy^{9}w^{8}-1918080xy^{7}w^{10}+2682720xy^{5}w^{12}+5541120xy^{3}w^{14}+9203712xyw^{16}-629856y^{16}w^{2}-1504656y^{14}w^{4}+1178064y^{12}w^{6}+2340576y^{10}w^{8}-979776y^{8}w^{10}-937872y^{6}w^{12}-1170288y^{4}w^{14}-2078976y^{2}w^{16}+z^{18}+5z^{17}w-5z^{16}w^{2}-35z^{15}w^{3}+43z^{14}w^{4}-3z^{13}w^{5}-437z^{12}w^{6}+365z^{11}w^{7}-819z^{10}w^{8}-4329z^{9}w^{9}+857z^{8}w^{10}-21009z^{7}w^{11}-38527z^{6}w^{12}-64153z^{5}w^{13}-312863z^{4}w^{14}-526265z^{3}w^{15}+586246z^{2}w^{16}-478464zw^{17}+859392w^{18})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.2.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{3}Y^{2}-3X^{2}Y^{2}Z-X^{3}Z^{2}-9XY^{2}Z^{2}+X^{2}Z^{3}-3Y^{2}Z^{3}-XZ^{4}+Z^{5} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.96.2.e.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}zw+\frac{1}{2}w^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{4}yz^{3}w^{2}-\frac{3}{4}yz^{2}w^{3}-\frac{9}{4}yzw^{4}-\frac{3}{4}yw^{5}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -\frac{1}{2}zw-\frac{1}{2}w^{2}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-8.l.1.3 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.l.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-48.i.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.m.1.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.bo.1.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.cc.1.11 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.dj.1.12 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.dm.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.ds.1.10 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.dt.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.eq.1.6 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.er.1.11 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.ex.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.fa.1.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.ff.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.fg.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.fi.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.fl.1.10 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.7-48.k.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.o.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.x.2.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.z.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.bp.1.16 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2^{2}$ |
| 48.384.7-48.bs.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.by.2.15 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2^{2}$ |
| 48.384.7-48.bz.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.576.18-48.s.2.44 | $48$ | $3$ | $3$ | $18$ | $0$ | $1^{4}\cdot2^{2}\cdot8$ |
| 48.768.19-48.o.2.32 | $48$ | $4$ | $4$ | $19$ | $0$ | $1^{3}\cdot2^{3}\cdot8$ |
| 240.384.5-240.zl.2.23 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.zn.1.24 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.zt.1.11 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.zv.1.23 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bav.2.22 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bax.1.32 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bbl.1.27 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bbn.1.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bht.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bhv.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bif.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bih.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bka.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bkc.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bkh.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bkj.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.ki.2.29 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.kk.2.26 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.kp.2.29 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.kr.2.19 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.lx.2.24 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.lz.2.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.mj.2.29 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ml.2.23 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |