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Properties

Label 24.48.1.iq.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $1^{2}\cdot2\cdot3^{2}\cdot6\cdot8\cdot24$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.68

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&21\\4&19\end{bmatrix}$, $\begin{bmatrix}5&15\\16&17\end{bmatrix}$, $\begin{bmatrix}13&3\\4&1\end{bmatrix}$, $\begin{bmatrix}17&3\\20&5\end{bmatrix}$, $\begin{bmatrix}23&0\\12&5\end{bmatrix}$, $\begin{bmatrix}23&21\\12&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.iq.1.1, 24.96.1-24.iq.1.2, 24.96.1-24.iq.1.3, 24.96.1-24.iq.1.4, 24.96.1-24.iq.1.5, 24.96.1-24.iq.1.6, 24.96.1-24.iq.1.7, 24.96.1-24.iq.1.8, 24.96.1-24.iq.1.9, 24.96.1-24.iq.1.10, 24.96.1-24.iq.1.11, 24.96.1-24.iq.1.12, 24.96.1-24.iq.1.13, 24.96.1-24.iq.1.14, 24.96.1-24.iq.1.15, 24.96.1-24.iq.1.16, 24.96.1-24.iq.1.17, 24.96.1-24.iq.1.18, 24.96.1-24.iq.1.19, 24.96.1-24.iq.1.20, 24.96.1-24.iq.1.21, 24.96.1-24.iq.1.22, 24.96.1-24.iq.1.23, 24.96.1-24.iq.1.24, 24.96.1-24.iq.1.25, 24.96.1-24.iq.1.26, 24.96.1-24.iq.1.27, 24.96.1-24.iq.1.28, 24.96.1-24.iq.1.29, 24.96.1-24.iq.1.30, 24.96.1-24.iq.1.31, 24.96.1-24.iq.1.32, 120.96.1-24.iq.1.1, 120.96.1-24.iq.1.2, 120.96.1-24.iq.1.3, 120.96.1-24.iq.1.4, 120.96.1-24.iq.1.5, 120.96.1-24.iq.1.6, 120.96.1-24.iq.1.7, 120.96.1-24.iq.1.8, 120.96.1-24.iq.1.9, 120.96.1-24.iq.1.10, 120.96.1-24.iq.1.11, 120.96.1-24.iq.1.12, 120.96.1-24.iq.1.13, 120.96.1-24.iq.1.14, 120.96.1-24.iq.1.15, 120.96.1-24.iq.1.16, 120.96.1-24.iq.1.17, 120.96.1-24.iq.1.18, 120.96.1-24.iq.1.19, 120.96.1-24.iq.1.20, 120.96.1-24.iq.1.21, 120.96.1-24.iq.1.22, 120.96.1-24.iq.1.23, 120.96.1-24.iq.1.24, 120.96.1-24.iq.1.25, 120.96.1-24.iq.1.26, 120.96.1-24.iq.1.27, 120.96.1-24.iq.1.28, 120.96.1-24.iq.1.29, 120.96.1-24.iq.1.30, 120.96.1-24.iq.1.31, 120.96.1-24.iq.1.32, 168.96.1-24.iq.1.1, 168.96.1-24.iq.1.2, 168.96.1-24.iq.1.3, 168.96.1-24.iq.1.4, 168.96.1-24.iq.1.5, 168.96.1-24.iq.1.6, 168.96.1-24.iq.1.7, 168.96.1-24.iq.1.8, 168.96.1-24.iq.1.9, 168.96.1-24.iq.1.10, 168.96.1-24.iq.1.11, 168.96.1-24.iq.1.12, 168.96.1-24.iq.1.13, 168.96.1-24.iq.1.14, 168.96.1-24.iq.1.15, 168.96.1-24.iq.1.16, 168.96.1-24.iq.1.17, 168.96.1-24.iq.1.18, 168.96.1-24.iq.1.19, 168.96.1-24.iq.1.20, 168.96.1-24.iq.1.21, 168.96.1-24.iq.1.22, 168.96.1-24.iq.1.23, 168.96.1-24.iq.1.24, 168.96.1-24.iq.1.25, 168.96.1-24.iq.1.26, 168.96.1-24.iq.1.27, 168.96.1-24.iq.1.28, 168.96.1-24.iq.1.29, 168.96.1-24.iq.1.30, 168.96.1-24.iq.1.31, 168.96.1-24.iq.1.32, 264.96.1-24.iq.1.1, 264.96.1-24.iq.1.2, 264.96.1-24.iq.1.3, 264.96.1-24.iq.1.4, 264.96.1-24.iq.1.5, 264.96.1-24.iq.1.6, 264.96.1-24.iq.1.7, 264.96.1-24.iq.1.8, 264.96.1-24.iq.1.9, 264.96.1-24.iq.1.10, 264.96.1-24.iq.1.11, 264.96.1-24.iq.1.12, 264.96.1-24.iq.1.13, 264.96.1-24.iq.1.14, 264.96.1-24.iq.1.15, 264.96.1-24.iq.1.16, 264.96.1-24.iq.1.17, 264.96.1-24.iq.1.18, 264.96.1-24.iq.1.19, 264.96.1-24.iq.1.20, 264.96.1-24.iq.1.21, 264.96.1-24.iq.1.22, 264.96.1-24.iq.1.23, 264.96.1-24.iq.1.24, 264.96.1-24.iq.1.25, 264.96.1-24.iq.1.26, 264.96.1-24.iq.1.27, 264.96.1-24.iq.1.28, 264.96.1-24.iq.1.29, 264.96.1-24.iq.1.30, 264.96.1-24.iq.1.31, 264.96.1-24.iq.1.32, 312.96.1-24.iq.1.1, 312.96.1-24.iq.1.2, 312.96.1-24.iq.1.3, 312.96.1-24.iq.1.4, 312.96.1-24.iq.1.5, 312.96.1-24.iq.1.6, 312.96.1-24.iq.1.7, 312.96.1-24.iq.1.8, 312.96.1-24.iq.1.9, 312.96.1-24.iq.1.10, 312.96.1-24.iq.1.11, 312.96.1-24.iq.1.12, 312.96.1-24.iq.1.13, 312.96.1-24.iq.1.14, 312.96.1-24.iq.1.15, 312.96.1-24.iq.1.16, 312.96.1-24.iq.1.17, 312.96.1-24.iq.1.18, 312.96.1-24.iq.1.19, 312.96.1-24.iq.1.20, 312.96.1-24.iq.1.21, 312.96.1-24.iq.1.22, 312.96.1-24.iq.1.23, 312.96.1-24.iq.1.24, 312.96.1-24.iq.1.25, 312.96.1-24.iq.1.26, 312.96.1-24.iq.1.27, 312.96.1-24.iq.1.28, 312.96.1-24.iq.1.29, 312.96.1-24.iq.1.30, 312.96.1-24.iq.1.31, 312.96.1-24.iq.1.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} - 17x - 15 $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(5:0:1)$, $(-1:0:1)$, $(-3:0:1)$

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4}\cdot\frac{32x^{2}y^{14}+49440x^{2}y^{12}z^{2}+41204736x^{2}y^{10}z^{4}+17975260160x^{2}y^{8}z^{6}+4069732843520x^{2}y^{6}z^{8}+499381945565184x^{2}y^{4}z^{10}+27895158454353920x^{2}y^{2}z^{12}+554111127877844992x^{2}z^{14}+464xy^{14}z+559680xy^{12}z^{3}+325064448xy^{10}z^{5}+115976615936xy^{8}z^{7}+23689694150656xy^{6}z^{9}+2510607340929024xy^{4}z^{11}+123361399543955456xy^{2}z^{13}+2219694395765030912xz^{15}+y^{16}+4464y^{14}z^{2}+4953120y^{12}z^{4}+2282817280y^{10}z^{6}+583111623680y^{8}z^{8}+88557304872960y^{6}z^{10}+6321869422133248y^{4}z^{12}+188088919423713280y^{2}z^{14}+1672083105113964544z^{16}}{z^{2}y^{2}(x^{2}y^{10}+2912x^{2}y^{8}z^{2}+550592x^{2}y^{6}z^{4}+22712320x^{2}y^{4}z^{6}-524288x^{2}y^{2}z^{8}+2097152x^{2}z^{10}+26xy^{10}z+20336xy^{8}z^{3}+2680448xy^{6}z^{5}+90521600xy^{4}z^{7}+2228224xy^{2}z^{9}-8388608xz^{11}+321y^{10}z^{2}+96208y^{8}z^{4}+5896128y^{6}z^{6}+68038656y^{4}z^{8}+1703936y^{2}z^{10}-10485760z^{12})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.m.1 $8$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.m.1 $8$ $4$ $4$ $0$ $0$ full Jacobian
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ 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
24.96.1.dd.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dd.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dd.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dd.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.df.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.df.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.df.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.df.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.ck.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.co.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.de.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.dg.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.ey.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.fa.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.fc.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.fe.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gk.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gk.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gk.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gk.4 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gm.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gm.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gm.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gm.4 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.ei.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.bk.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.cy.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.dg.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.sd.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sd.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sd.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sd.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sf.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sf.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sf.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sf.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ny.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.oa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.oc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.oe.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.po.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sa.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sa.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sa.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sc.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sc.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sc.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.bra.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkge.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.sb.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sb.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sb.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sb.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sd.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sd.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sd.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sd.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.lo.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.lq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ls.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.lu.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mu.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mw.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.my.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.na.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pm.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pm.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pm.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.po.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.po.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.po.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.po.4 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.sb.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sb.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sb.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sb.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sd.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sd.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sd.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sd.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.lo.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.lq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ls.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.lu.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mu.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mw.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.my.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.na.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pm.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pm.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pm.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.po.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.po.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.po.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.po.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.sd.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sd.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sd.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sd.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sf.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sf.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sf.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sf.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ny.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.oa.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.oc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.oe.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pi.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pk.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pm.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.po.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sa.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sa.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sa.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sa.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sc.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sc.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sc.4 $312$ $2$ $2$ $3$ $?$ not computed