Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $144$ | ||
| 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 | $4^{8}\cdot8^{4}\cdot16^{2}$ | 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: | 16I2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.2.4 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&6\\24&7\end{bmatrix}$, $\begin{bmatrix}1&12\\32&37\end{bmatrix}$, $\begin{bmatrix}25&6\\40&43\end{bmatrix}$, $\begin{bmatrix}37&28\\36&29\end{bmatrix}$, $\begin{bmatrix}41&2\\24&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.2.d.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $16$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{8}\cdot3^{4}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 144.2.k.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z w - y z w + y w^{2} $ |
| $=$ | $2 x z^{2} - y z^{2} + y z w$ | |
| $=$ | $2 x y z - y^{2} z + y^{2} w$ | |
| $=$ | $2 x^{2} z - x y z + x y w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} - x^{4} z + 3 x^{3} y^{2} - 3 x^{3} z^{2} - 3 x^{2} y^{2} z - x^{2} z^{3} + 3 x y^{2} z^{2} - 3 y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x^{2} + x + 1\right) y $ | $=$ | $ -x^{6} + x^{5} - x^{3} - 2x - 1 $ |
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 |
|---|
| $(0:0:-1:1)$, $(0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{7737780960x^{2}y^{18}-20641640832x^{2}y^{16}w^{2}+11172595680x^{2}y^{14}w^{4}-2310031872x^{2}y^{12}w^{6}-4734044352x^{2}y^{10}w^{8}-7013765376x^{2}y^{8}w^{10}-10399181760x^{2}y^{6}w^{12}-12421513728x^{2}y^{4}w^{14}-8477708832x^{2}y^{2}w^{16}+13375966848x^{2}w^{18}+1889568xy^{19}-12897561312xy^{17}w^{2}+30104177472xy^{15}w^{4}-13746817152xy^{13}w^{6}+4627295424xy^{11}w^{8}+3955355712xy^{9}w^{10}+3537779328xy^{7}w^{12}-1419686784xy^{5}w^{14}-16944147936xy^{3}w^{16}-54590921568xyw^{18}-944784y^{20}-629856y^{18}w^{2}+3437649072y^{16}w^{4}-6884745984y^{14}w^{6}+1128958560y^{12}w^{8}-1317083328y^{10}w^{10}-726125472y^{8}w^{12}+1109852928y^{6}w^{14}+5234037552y^{4}w^{16}+12392890656y^{2}w^{18}-32760z^{20}+65535z^{19}w+687790z^{18}w^{2}-459077z^{17}w^{3}-7169904z^{16}w^{4}-4969736z^{15}w^{5}+35939232z^{14}w^{6}+77651664z^{13}w^{7}-13931144z^{12}w^{8}-314511010z^{11}w^{9}-699221772z^{10}w^{10}-373735554z^{9}w^{11}+2558849432z^{8}w^{12}+6646333296z^{7}w^{13}+902192136z^{6}w^{14}-18263072088z^{5}w^{15}-20577975168z^{4}w^{16}+12536808491z^{3}w^{17}+28317924526z^{2}w^{18}+11327814735zw^{19}+1114663904w^{20}}{w^{5}(23328x^{2}y^{10}w^{3}+93312x^{2}y^{8}w^{5}+261792x^{2}y^{6}w^{7}+622080x^{2}y^{4}w^{9}+1296000x^{2}y^{2}w^{11}+2400768x^{2}w^{13}-23328xy^{11}w^{3}-85536xy^{9}w^{5}-222912xy^{7}w^{7}-490752xy^{5}w^{9}-930816xy^{3}w^{11}-1497216xyw^{13}+11664y^{12}w^{3}+38880y^{10}w^{5}+94608y^{8}w^{7}+196992y^{6}w^{9}+353664y^{4}w^{11}+534912y^{2}w^{13}+z^{15}+10z^{14}w+25z^{13}w^{2}-88z^{12}w^{3}-554z^{11}w^{4}-524z^{10}w^{5}+1990z^{9}w^{6}+4408z^{8}w^{7}+5513z^{7}w^{8}+24674z^{6}w^{9}+13185z^{5}w^{10}-204192z^{4}w^{11}-354752z^{3}w^{12}+290720z^{2}w^{13}+649664zw^{14}+200064w^{15})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}+3X^{3}Y^{2}-X^{4}Z-3X^{2}Y^{2}Z-3X^{3}Z^{2}+3XY^{2}Z^{2}-X^{2}Z^{3}-3Y^{2}Z^{3} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.96.2.d.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{3}{2}yz^{5}+\frac{3}{2}yz^{4}w-\frac{3}{2}yz^{3}w^{2}+\frac{3}{2}yz^{2}w^{3}-\frac{1}{2}z^{6}-\frac{1}{2}z^{5}w-\frac{1}{2}z^{4}w^{2}-\frac{1}{2}z^{3}w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle zw$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.c.1.1 | $48$ | $2$ | $2$ | $0$ | $0$ | 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.p.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.p.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.t.1.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.t.2.5 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.bo.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.bo.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.bs.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.bs.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.cx.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.cx.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.cz.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.cz.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.dd.1.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.dd.2.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1\cdot2$ |
| 48.384.5-48.dh.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.5-48.dh.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1\cdot2$ |
| 48.384.7-48.q.1.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2^{2}$ |
| 48.384.7-48.s.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.v.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.y.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.bh.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.bi.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.bm.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2^{2}$ |
| 48.384.7-48.bo.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2^{2}$ |
| 48.576.18-48.n.1.7 | $48$ | $3$ | $3$ | $18$ | $0$ | $1^{4}\cdot2^{2}\cdot8$ |
| 48.768.19-48.j.1.1 | $48$ | $4$ | $4$ | $19$ | $0$ | $1^{3}\cdot2^{3}\cdot8$ |
| 240.384.5-240.ny.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ny.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oa.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oa.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.og.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.og.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oi.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.oi.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qu.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qu.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qw.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.qw.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.rc.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.rc.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.re.1.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.re.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.fn.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fo.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fr.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.fs.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hg.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hh.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hk.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.hl.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |