Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $48$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $6$ are rational) | Cusp widths | $1^{4}\cdot3^{4}\cdot4\cdot12\cdot16\cdot48$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48I3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.4493 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&15\\12&31\end{bmatrix}$, $\begin{bmatrix}7&47\\0&41\end{bmatrix}$, $\begin{bmatrix}13&5\\12&37\end{bmatrix}$, $\begin{bmatrix}25&44\\0&41\end{bmatrix}$, $\begin{bmatrix}31&1\\36&47\end{bmatrix}$, $\begin{bmatrix}37&5\\36&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.3.pv.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $8$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{11}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 48.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x w t + y t^{2} $ |
| $=$ | $x w^{2} + y w t$ | |
| $=$ | $x z w + y z t$ | |
| $=$ | $x^{2} w + x y t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} - x^{5} y - x^{4} y^{2} + x^{4} z^{2} + 2 x^{3} y z^{2} - 4 x^{2} z^{4} - x y z^{4} + z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -3x^{4} + 2 $ |
Rational points
This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(1:0:0:0:0)$, $(0:-2:1:1:0)$, $(0:1:-2:1:0)$, $(0:0:1:0:0)$, $(0:0:0:1:1)$, $(0:0:0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{8x^{13}t-4x^{12}z^{2}+168x^{12}t^{2}-88x^{11}z^{2}t+1504x^{11}t^{3}-840x^{10}z^{2}t^{2}+7492x^{10}t^{4}-4593x^{9}z^{2}t^{3}+23126x^{9}t^{5}-15967x^{8}z^{2}t^{4}+45258x^{8}t^{6}-39079x^{7}z^{2}t^{5}+50962x^{7}t^{7}-68051x^{6}z^{2}t^{6}+19678x^{6}t^{8}-74976x^{5}z^{2}t^{7}-21272x^{5}t^{9}-51214x^{4}z^{2}t^{8}-49300x^{4}t^{10}-40894x^{3}z^{2}t^{9}-57828x^{3}t^{11}-39086x^{2}z^{2}t^{10}+142052x^{2}t^{12}-114088xz^{2}t^{11}+465696xt^{13}+398568yzt^{12}-z^{14}-42z^{12}t^{2}-682z^{10}t^{4}-5396z^{8}t^{6}-21542z^{6}t^{8}-40484z^{4}t^{10}-221484z^{2}t^{12}-6190zw^{13}-19096zw^{11}t^{2}-21796zw^{9}t^{4}+31864zw^{7}t^{6}+653978zw^{5}t^{8}+79600zw^{3}t^{10}-1183736zwt^{12}+6920w^{14}+25342w^{12}t^{2}-28554w^{10}t^{4}-21092w^{8}t^{6}+1300352w^{6}t^{8}-443394w^{4}t^{10}-1539590w^{2}t^{12}+700144t^{14}}{t^{4}(3xt^{9}+5yzt^{8}+z^{2}t^{8}-27zw^{9}+54zw^{5}t^{4}-30zwt^{8}-54w^{10}+27w^{8}t^{2}+117w^{6}t^{4}-54w^{4}t^{6}-68w^{2}t^{8}+32t^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.pv.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}-X^{5}Y-X^{4}Y^{2}+X^{4}Z^{2}+2X^{3}YZ^{2}-4X^{2}Z^{4}-XYZ^{4}+Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.96.3.pv.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -zw^{3}-w^{4}+w^{2}t^{2}-t^{4}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(3)$ | $3$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
| 16.24.0-8.n.1.8 | $16$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.ir.1.46 | $24$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.kq.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.kq.3.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.ks.1.13 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.384.5-48.ks.3.10 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 48.384.5-48.ky.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.ky.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.la.1.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.5-48.la.2.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 48.384.7-48.ep.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ep.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eq.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.eq.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.er.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.er.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.es.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.es.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ex.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ex.3.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ey.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ey.3.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ez.2.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ez.4.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fa.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fa.4.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fb.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fb.4.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fc.2.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fc.4.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fd.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fd.3.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.3.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fn.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fn.3.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fo.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fo.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fp.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fp.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fq.1.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fq.3.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.9-48.hp.2.45 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.ix.1.22 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.mi.1.37 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.mo.1.22 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.baf.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.baf.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bah.1.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bah.2.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bef.2.25 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.beh.2.26 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot4$ |
| 48.384.9-48.bej.2.25 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot4$ |
| 48.384.9-48.bel.2.26 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot4$ |
| 48.384.9-48.bev.1.13 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bev.3.10 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bex.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bex.3.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.576.13-48.cp.1.5 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 240.384.5-240.cka.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cka.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckb.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckb.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckq.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckq.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckr.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ckr.2.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-240.qr.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qr.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qs.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qs.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qt.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qt.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qu.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qu.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qz.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.qz.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ra.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ra.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rb.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rb.3.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rc.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rc.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rd.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rd.3.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.re.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.re.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rf.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rf.2.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rg.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rg.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rp.1.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rp.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rq.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rq.3.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rr.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rr.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rs.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.rs.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.foz.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpa.1.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpd.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpe.1.26 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpp.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpp.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpq.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fpq.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.frl.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.frm.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.frp.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.frq.2.50 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsb.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsb.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsc.1.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsc.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |