Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $72$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $8$ are rational) | Cusp widths | $3^{8}\cdot6^{4}\cdot24^{4}$ | Cusp orbits | $1^{8}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24R5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.288.5.2 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}11&3\\12&43\end{bmatrix}$, $\begin{bmatrix}13&45\\12&13\end{bmatrix}$, $\begin{bmatrix}35&9\\0&1\end{bmatrix}$, $\begin{bmatrix}35&21\\24&37\end{bmatrix}$, $\begin{bmatrix}37&42\\12&47\end{bmatrix}$, $\begin{bmatrix}43&30\\0&19\end{bmatrix}$, $\begin{bmatrix}47&39\\36&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.144.5.ej.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $4096$ |
Jacobian
| Conductor: | $2^{13}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 24.2.a.a$^{2}$, 36.2.a.a$^{2}$, 72.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - y w - z w $ |
| $=$ | $x^{2} - y z + y w - y t + z^{2} + z t$ | |
| $=$ | $x^{2} - 2 y z + y t - 2 z^{2} + 2 z w - 3 z t - w^{2} + w t - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} y^{2} + x^{4} z^{2} - 4 x^{2} y^{2} z^{2} + y^{4} z^{2} - y^{2} z^{4} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:-1:0:1)$, $(-1/2:0:-1/2:-1/2:1)$, $(0:1:0:0:0)$, $(0:-1/2:1/2:1:0)$, $(0:1/2:-1/2:1:1)$, $(1/2:1/2:0:1/2:1)$, $(-1/2:1/2:0:1/2:1)$, $(1/2:0:-1/2:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{262144y^{18}-1572864y^{15}t^{3}+9437184y^{14}t^{4}-47185920y^{13}t^{5}+216268800y^{12}t^{6}-943718400y^{11}t^{7}+3982491648y^{10}t^{8}-16355164160y^{9}t^{9}+65569554432y^{8}t^{10}-257182138368y^{7}t^{11}+989471440896y^{6}t^{12}-3750469042176y^{5}t^{13}+14114403385344y^{4}t^{14}-53434856767488y^{3}t^{15}+207527309475840y^{2}t^{16}-63yw^{17}+1178487yw^{16}t-7385481yw^{15}t^{2}-134916669yw^{14}t^{3}+2046485448yw^{13}t^{4}-10055049282yw^{12}t^{5}+4732047135yw^{11}t^{6}+137159242839yw^{10}t^{7}-368339355693yw^{9}t^{8}-1018609882209yw^{8}t^{9}+4843162091583yw^{7}t^{10}+2775071866779yw^{6}t^{11}-44614419555186yw^{5}t^{12}-9937070076024yw^{4}t^{13}+112004583468579yw^{3}t^{14}-379917969199101yw^{2}t^{15}-573941649919239ywt^{16}+43340662929681yt^{17}-63zw^{17}+3537783zw^{16}t-58700043zw^{15}t^{2}+333406161zw^{14}t^{3}+128555262zw^{13}t^{4}-10207186056zw^{12}t^{5}+38064970089zw^{11}t^{6}+43330945977zw^{10}t^{7}-571381327863zw^{9}t^{8}+352922366853zw^{8}t^{9}+4953010212333zw^{7}t^{10}-9214659241911zw^{6}t^{11}-31706143302636zw^{5}t^{12}+52596777575322zw^{4}t^{13}+3393221177637zw^{3}t^{14}-418042201619907zw^{2}t^{15}+115336882673235zwt^{16}+58428809115759zt^{17}+63w^{18}-1178613w^{17}t+25667640w^{16}t^{2}-215760033w^{15}t^{3}+653141763w^{14}t^{4}+2024764110w^{13}t^{5}-19678930443w^{12}t^{6}+24828240087w^{11}t^{7}+200659091868w^{10}t^{8}-541874982159w^{9}t^{9}-1151409877362w^{8}t^{10}+6403512411495w^{7}t^{11}+5808785268573w^{6}t^{12}-24224925757572w^{5}t^{13}+48101848056207w^{4}t^{14}+153288044133939w^{3}t^{15}-111577379898294w^{2}t^{16}+130425033577905wt^{17}+58428809377903t^{18}}{t^{3}(4096y^{6}t^{9}-98304y^{5}t^{10}+1376256y^{4}t^{11}-14704640y^{3}t^{12}+132857856y^{2}t^{13}+511yw^{14}-7190yw^{13}t+35173yw^{12}t^{2}-27391yw^{11}t^{3}-407587yw^{10}t^{4}+886922yw^{9}t^{5}+376283yw^{8}t^{6}-6972208yw^{7}t^{7}-25292173yw^{6}t^{8}-60166592yw^{5}t^{9}-118179605yw^{4}t^{10}-206301887yw^{3}t^{11}-356036544yw^{2}t^{12}-328607424ywt^{13}+31201344yt^{14}+511zw^{14}-8726zw^{13}t+61799zw^{12}t^{2}-198223zw^{11}t^{3}+41579zw^{10}t^{4}+848152zw^{9}t^{5}-1127725zw^{8}t^{6}-7159588zw^{7}t^{7}-18951011zw^{6}t^{8}-38932130zw^{5}t^{9}-69556013zw^{4}t^{10}-115593025zw^{3}t^{11}-191195712zw^{2}t^{12}+91050432zwt^{13}+35049408zt^{14}+w^{15}+1300w^{14}t-17938w^{13}t^{2}+95094w^{12}t^{3}-135079w^{11}t^{4}-164681w^{10}t^{5}+1042615w^{9}t^{6}+3522188w^{8}t^{7}+8913052w^{7}t^{8}+19019583w^{6}t^{9}+35694793w^{5}t^{10}+62378129w^{4}t^{11}+94701631w^{3}t^{12}-9543936w^{2}t^{13}+94898496wt^{14}+35049408t^{15})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.5.ej.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y-z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}Y^{2}+X^{4}Z^{2}-4X^{2}Y^{2}Z^{2}+Y^{4}Z^{2}-Y^{2}Z^{4} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 16.24.0-8.n.1.8 | $16$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.1-24.ir.1.17 | $48$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 48.144.3-24.or.1.8 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.144.3-24.or.1.23 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.576.9-24.p.1.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.576.9-24.p.2.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.576.9-24.r.1.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.576.9-24.r.2.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 48.576.13-48.bg.1.3 | $48$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 48.576.13-48.bg.1.11 | $48$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 48.576.13-48.bh.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-48.bh.1.6 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-48.bw.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.bw.1.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.bw.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.bw.2.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.bx.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bx.1.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bx.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bx.2.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.by.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.by.1.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.by.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.by.2.9 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bz.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bz.1.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bz.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.bz.2.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.ca.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.ca.1.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.ca.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.ca.2.9 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.cf.1.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.cf.1.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.cf.2.1 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.cf.2.17 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-48.co.1.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.co.1.21 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.cp.1.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.cp.1.21 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-48.cs.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-48.cs.1.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-48.cu.1.3 | $48$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 48.576.13-48.cu.1.19 | $48$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 48.576.13-24.ec.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-24.et.1.10 | $48$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 48.576.13-24.go.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-24.gq.1.10 | $48$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 48.576.13-24.kv.1.14 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.kv.2.14 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.kx.1.14 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.kx.2.14 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lk.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lk.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lk.2.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lk.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.ll.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ll.1.18 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ll.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ll.2.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lm.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lm.1.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lm.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lm.2.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ln.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ln.1.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ln.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.ln.2.10 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lo.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lo.1.18 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lo.2.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lo.2.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 48.576.13-24.lp.1.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lp.1.18 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lp.2.2 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.lp.2.18 | $48$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4$ |
| 48.576.13-24.md.1.3 | $48$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 48.576.13-24.mg.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.13-24.mm.1.4 | $48$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 48.576.13-24.ms.1.4 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 48.576.17-48.ze.1.1 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |
| 48.576.17-48.ze.1.33 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |
| 48.576.17-48.zg.1.1 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{12}$ |
| 48.576.17-48.zg.1.33 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{12}$ |
| 48.576.17-48.caa.1.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 48.576.17-48.caa.1.33 | $48$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 48.576.17-48.cab.1.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 48.576.17-48.cab.1.33 | $48$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 48.576.17-48.cae.1.1 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |
| 48.576.17-48.cae.1.5 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |
| 48.576.17-48.caf.1.1 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{12}$ |
| 48.576.17-48.caf.1.9 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{12}$ |