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Properties

Label 24.48.1.iv.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: $576$
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: 24.48.1.500

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&6\\8&13\end{bmatrix}$, $\begin{bmatrix}11&21\\20&1\end{bmatrix}$, $\begin{bmatrix}17&6\\8&13\end{bmatrix}$, $\begin{bmatrix}17&6\\8&17\end{bmatrix}$, $\begin{bmatrix}19&3\\20&7\end{bmatrix}$, $\begin{bmatrix}23&12\\4&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.iv.1.1, 24.96.1-24.iv.1.2, 24.96.1-24.iv.1.3, 24.96.1-24.iv.1.4, 24.96.1-24.iv.1.5, 24.96.1-24.iv.1.6, 24.96.1-24.iv.1.7, 24.96.1-24.iv.1.8, 24.96.1-24.iv.1.9, 24.96.1-24.iv.1.10, 24.96.1-24.iv.1.11, 24.96.1-24.iv.1.12, 24.96.1-24.iv.1.13, 24.96.1-24.iv.1.14, 24.96.1-24.iv.1.15, 24.96.1-24.iv.1.16, 24.96.1-24.iv.1.17, 24.96.1-24.iv.1.18, 24.96.1-24.iv.1.19, 24.96.1-24.iv.1.20, 24.96.1-24.iv.1.21, 24.96.1-24.iv.1.22, 24.96.1-24.iv.1.23, 24.96.1-24.iv.1.24, 24.96.1-24.iv.1.25, 24.96.1-24.iv.1.26, 24.96.1-24.iv.1.27, 24.96.1-24.iv.1.28, 24.96.1-24.iv.1.29, 24.96.1-24.iv.1.30, 24.96.1-24.iv.1.31, 24.96.1-24.iv.1.32, 120.96.1-24.iv.1.1, 120.96.1-24.iv.1.2, 120.96.1-24.iv.1.3, 120.96.1-24.iv.1.4, 120.96.1-24.iv.1.5, 120.96.1-24.iv.1.6, 120.96.1-24.iv.1.7, 120.96.1-24.iv.1.8, 120.96.1-24.iv.1.9, 120.96.1-24.iv.1.10, 120.96.1-24.iv.1.11, 120.96.1-24.iv.1.12, 120.96.1-24.iv.1.13, 120.96.1-24.iv.1.14, 120.96.1-24.iv.1.15, 120.96.1-24.iv.1.16, 120.96.1-24.iv.1.17, 120.96.1-24.iv.1.18, 120.96.1-24.iv.1.19, 120.96.1-24.iv.1.20, 120.96.1-24.iv.1.21, 120.96.1-24.iv.1.22, 120.96.1-24.iv.1.23, 120.96.1-24.iv.1.24, 120.96.1-24.iv.1.25, 120.96.1-24.iv.1.26, 120.96.1-24.iv.1.27, 120.96.1-24.iv.1.28, 120.96.1-24.iv.1.29, 120.96.1-24.iv.1.30, 120.96.1-24.iv.1.31, 120.96.1-24.iv.1.32, 168.96.1-24.iv.1.1, 168.96.1-24.iv.1.2, 168.96.1-24.iv.1.3, 168.96.1-24.iv.1.4, 168.96.1-24.iv.1.5, 168.96.1-24.iv.1.6, 168.96.1-24.iv.1.7, 168.96.1-24.iv.1.8, 168.96.1-24.iv.1.9, 168.96.1-24.iv.1.10, 168.96.1-24.iv.1.11, 168.96.1-24.iv.1.12, 168.96.1-24.iv.1.13, 168.96.1-24.iv.1.14, 168.96.1-24.iv.1.15, 168.96.1-24.iv.1.16, 168.96.1-24.iv.1.17, 168.96.1-24.iv.1.18, 168.96.1-24.iv.1.19, 168.96.1-24.iv.1.20, 168.96.1-24.iv.1.21, 168.96.1-24.iv.1.22, 168.96.1-24.iv.1.23, 168.96.1-24.iv.1.24, 168.96.1-24.iv.1.25, 168.96.1-24.iv.1.26, 168.96.1-24.iv.1.27, 168.96.1-24.iv.1.28, 168.96.1-24.iv.1.29, 168.96.1-24.iv.1.30, 168.96.1-24.iv.1.31, 168.96.1-24.iv.1.32, 264.96.1-24.iv.1.1, 264.96.1-24.iv.1.2, 264.96.1-24.iv.1.3, 264.96.1-24.iv.1.4, 264.96.1-24.iv.1.5, 264.96.1-24.iv.1.6, 264.96.1-24.iv.1.7, 264.96.1-24.iv.1.8, 264.96.1-24.iv.1.9, 264.96.1-24.iv.1.10, 264.96.1-24.iv.1.11, 264.96.1-24.iv.1.12, 264.96.1-24.iv.1.13, 264.96.1-24.iv.1.14, 264.96.1-24.iv.1.15, 264.96.1-24.iv.1.16, 264.96.1-24.iv.1.17, 264.96.1-24.iv.1.18, 264.96.1-24.iv.1.19, 264.96.1-24.iv.1.20, 264.96.1-24.iv.1.21, 264.96.1-24.iv.1.22, 264.96.1-24.iv.1.23, 264.96.1-24.iv.1.24, 264.96.1-24.iv.1.25, 264.96.1-24.iv.1.26, 264.96.1-24.iv.1.27, 264.96.1-24.iv.1.28, 264.96.1-24.iv.1.29, 264.96.1-24.iv.1.30, 264.96.1-24.iv.1.31, 264.96.1-24.iv.1.32, 312.96.1-24.iv.1.1, 312.96.1-24.iv.1.2, 312.96.1-24.iv.1.3, 312.96.1-24.iv.1.4, 312.96.1-24.iv.1.5, 312.96.1-24.iv.1.6, 312.96.1-24.iv.1.7, 312.96.1-24.iv.1.8, 312.96.1-24.iv.1.9, 312.96.1-24.iv.1.10, 312.96.1-24.iv.1.11, 312.96.1-24.iv.1.12, 312.96.1-24.iv.1.13, 312.96.1-24.iv.1.14, 312.96.1-24.iv.1.15, 312.96.1-24.iv.1.16, 312.96.1-24.iv.1.17, 312.96.1-24.iv.1.18, 312.96.1-24.iv.1.19, 312.96.1-24.iv.1.20, 312.96.1-24.iv.1.21, 312.96.1-24.iv.1.22, 312.96.1-24.iv.1.23, 312.96.1-24.iv.1.24, 312.96.1-24.iv.1.25, 312.96.1-24.iv.1.26, 312.96.1-24.iv.1.27, 312.96.1-24.iv.1.28, 312.96.1-24.iv.1.29, 312.96.1-24.iv.1.30, 312.96.1-24.iv.1.31, 312.96.1-24.iv.1.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 156x - 560 $
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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
$(0:1:0)$, $(14:0:1)$, $(-4:0:1)$, $(-10:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4\cdot3^4}\cdot\frac{96x^{2}y^{14}+4004640x^{2}y^{12}z^{2}+90114757632x^{2}y^{10}z^{4}+1061421137187840x^{2}y^{8}z^{6}+6488468676279336960x^{2}y^{6}z^{8}+21496755283181662961664x^{2}y^{4}z^{10}+32421467787354839596400640x^{2}y^{2}z^{12}+17388594333944875364414128128x^{2}z^{14}+4368xy^{14}z+144011520xy^{12}z^{3}+2312977358592xy^{10}z^{5}+22667751857590272xy^{8}z^{7}+126284010094627651584xy^{6}z^{9}+367213751802934056714240xy^{4}z^{11}+494977539190102077012443136xy^{2}z^{13}+243746275109686767350427156480xz^{15}+y^{16}+124800y^{14}z^{2}+3750831360y^{12}z^{4}+47155555123200y^{10}z^{6}+331495755120525312y^{8}z^{8}+1390496073211098169344y^{6}z^{10}+2794938739436762757070848y^{4}z^{12}+2430032602145395360017678336y^{2}z^{14}+698603337110790098828201558016z^{16}}{z^{2}y^{2}(x^{2}y^{10}+78624x^{2}y^{8}z^{2}+401381568x^{2}y^{6}z^{4}+447046594560x^{2}y^{4}z^{6}-278628139008x^{2}y^{2}z^{8}+30091839012864x^{2}z^{10}+80xy^{10}z+1804464xy^{8}z^{3}+6664902912xy^{6}z^{5}+6239303147520xy^{4}z^{7}+2995252494336xy^{2}z^{9}-300918390128640xz^{11}+2968y^{10}z^{2}+25104384y^{8}z^{4}+44948017152y^{6}z^{6}+17845100347392y^{4}z^{8}+11423753699328y^{2}z^{10}-1685142984720384z^{12})}$

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

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.bb.1 $24$ $4$ $4$ $0$ $0$ 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.dm.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dm.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dm.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dm.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.do.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.do.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.do.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.do.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.cg.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.cq.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.en.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.eo.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.ey.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.fb.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.gb.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gc.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gt.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gt.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gt.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gt.4 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gv.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gv.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gv.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gv.4 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.eq.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.bp.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.dd.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.dl.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.tc.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tc.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tc.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tc.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.te.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.te.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.te.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.te.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ot.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ov.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pd.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sz.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sz.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sz.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tb.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tb.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tb.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.brn.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkgz.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.ta.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ta.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ta.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ta.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tc.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tc.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tc.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tc.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.mf.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mh.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mn.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mp.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nl.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nn.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nt.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nv.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ql.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ql.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ql.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ql.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qn.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qn.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qn.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qn.4 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.ta.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ta.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ta.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ta.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tc.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tc.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tc.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tc.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.mf.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mh.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mn.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mp.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nl.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nn.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nt.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nv.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ql.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ql.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ql.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ql.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qn.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qn.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qn.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qn.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.tc.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tc.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tc.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tc.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.te.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.te.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.te.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.te.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ot.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ov.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pb.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pd.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qb.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qh.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qj.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sz.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sz.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sz.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tb.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tb.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tb.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tb.4 $312$ $2$ $2$ $3$ $?$ not computed