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Properties

Label 24.72.4.fd.1
Level $24$
Index $72$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

Related objects

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $144$
Index: $72$ $\PSL_2$-index:$72$
Genus: $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $6^{4}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24D4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.72.4.235

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&20\\8&9\end{bmatrix}$, $\begin{bmatrix}5&6\\0&7\end{bmatrix}$, $\begin{bmatrix}7&20\\8&23\end{bmatrix}$, $\begin{bmatrix}11&21\\0&7\end{bmatrix}$, $\begin{bmatrix}13&10\\16&17\end{bmatrix}$, $\begin{bmatrix}13&20\\16&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.144.4-24.fd.1.1, 24.144.4-24.fd.1.2, 24.144.4-24.fd.1.3, 24.144.4-24.fd.1.4, 24.144.4-24.fd.1.5, 24.144.4-24.fd.1.6, 24.144.4-24.fd.1.7, 24.144.4-24.fd.1.8, 24.144.4-24.fd.1.9, 24.144.4-24.fd.1.10, 24.144.4-24.fd.1.11, 24.144.4-24.fd.1.12, 24.144.4-24.fd.1.13, 24.144.4-24.fd.1.14, 24.144.4-24.fd.1.15, 24.144.4-24.fd.1.16, 24.144.4-24.fd.1.17, 24.144.4-24.fd.1.18, 24.144.4-24.fd.1.19, 24.144.4-24.fd.1.20, 24.144.4-24.fd.1.21, 24.144.4-24.fd.1.22, 24.144.4-24.fd.1.23, 24.144.4-24.fd.1.24, 48.144.4-24.fd.1.1, 48.144.4-24.fd.1.2, 48.144.4-24.fd.1.3, 48.144.4-24.fd.1.4, 48.144.4-24.fd.1.5, 48.144.4-24.fd.1.6, 48.144.4-24.fd.1.7, 48.144.4-24.fd.1.8, 48.144.4-24.fd.1.9, 48.144.4-24.fd.1.10, 48.144.4-24.fd.1.11, 48.144.4-24.fd.1.12, 48.144.4-24.fd.1.13, 48.144.4-24.fd.1.14, 48.144.4-24.fd.1.15, 48.144.4-24.fd.1.16, 120.144.4-24.fd.1.1, 120.144.4-24.fd.1.2, 120.144.4-24.fd.1.3, 120.144.4-24.fd.1.4, 120.144.4-24.fd.1.5, 120.144.4-24.fd.1.6, 120.144.4-24.fd.1.7, 120.144.4-24.fd.1.8, 120.144.4-24.fd.1.9, 120.144.4-24.fd.1.10, 120.144.4-24.fd.1.11, 120.144.4-24.fd.1.12, 120.144.4-24.fd.1.13, 120.144.4-24.fd.1.14, 120.144.4-24.fd.1.15, 120.144.4-24.fd.1.16, 120.144.4-24.fd.1.17, 120.144.4-24.fd.1.18, 120.144.4-24.fd.1.19, 120.144.4-24.fd.1.20, 120.144.4-24.fd.1.21, 120.144.4-24.fd.1.22, 120.144.4-24.fd.1.23, 120.144.4-24.fd.1.24, 168.144.4-24.fd.1.1, 168.144.4-24.fd.1.2, 168.144.4-24.fd.1.3, 168.144.4-24.fd.1.4, 168.144.4-24.fd.1.5, 168.144.4-24.fd.1.6, 168.144.4-24.fd.1.7, 168.144.4-24.fd.1.8, 168.144.4-24.fd.1.9, 168.144.4-24.fd.1.10, 168.144.4-24.fd.1.11, 168.144.4-24.fd.1.12, 168.144.4-24.fd.1.13, 168.144.4-24.fd.1.14, 168.144.4-24.fd.1.15, 168.144.4-24.fd.1.16, 168.144.4-24.fd.1.17, 168.144.4-24.fd.1.18, 168.144.4-24.fd.1.19, 168.144.4-24.fd.1.20, 168.144.4-24.fd.1.21, 168.144.4-24.fd.1.22, 168.144.4-24.fd.1.23, 168.144.4-24.fd.1.24, 240.144.4-24.fd.1.1, 240.144.4-24.fd.1.2, 240.144.4-24.fd.1.3, 240.144.4-24.fd.1.4, 240.144.4-24.fd.1.5, 240.144.4-24.fd.1.6, 240.144.4-24.fd.1.7, 240.144.4-24.fd.1.8, 240.144.4-24.fd.1.9, 240.144.4-24.fd.1.10, 240.144.4-24.fd.1.11, 240.144.4-24.fd.1.12, 240.144.4-24.fd.1.13, 240.144.4-24.fd.1.14, 240.144.4-24.fd.1.15, 240.144.4-24.fd.1.16, 264.144.4-24.fd.1.1, 264.144.4-24.fd.1.2, 264.144.4-24.fd.1.3, 264.144.4-24.fd.1.4, 264.144.4-24.fd.1.5, 264.144.4-24.fd.1.6, 264.144.4-24.fd.1.7, 264.144.4-24.fd.1.8, 264.144.4-24.fd.1.9, 264.144.4-24.fd.1.10, 264.144.4-24.fd.1.11, 264.144.4-24.fd.1.12, 264.144.4-24.fd.1.13, 264.144.4-24.fd.1.14, 264.144.4-24.fd.1.15, 264.144.4-24.fd.1.16, 264.144.4-24.fd.1.17, 264.144.4-24.fd.1.18, 264.144.4-24.fd.1.19, 264.144.4-24.fd.1.20, 264.144.4-24.fd.1.21, 264.144.4-24.fd.1.22, 264.144.4-24.fd.1.23, 264.144.4-24.fd.1.24, 312.144.4-24.fd.1.1, 312.144.4-24.fd.1.2, 312.144.4-24.fd.1.3, 312.144.4-24.fd.1.4, 312.144.4-24.fd.1.5, 312.144.4-24.fd.1.6, 312.144.4-24.fd.1.7, 312.144.4-24.fd.1.8, 312.144.4-24.fd.1.9, 312.144.4-24.fd.1.10, 312.144.4-24.fd.1.11, 312.144.4-24.fd.1.12, 312.144.4-24.fd.1.13, 312.144.4-24.fd.1.14, 312.144.4-24.fd.1.15, 312.144.4-24.fd.1.16, 312.144.4-24.fd.1.17, 312.144.4-24.fd.1.18, 312.144.4-24.fd.1.19, 312.144.4-24.fd.1.20, 312.144.4-24.fd.1.21, 312.144.4-24.fd.1.22, 312.144.4-24.fd.1.23, 312.144.4-24.fd.1.24
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1024$

Jacobian

Conductor: $2^{10}\cdot3^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 36.2.a.a$^{3}$, 144.2.a.a

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 15 y^{2} - 2 y w + 3 z^{2} - w^{2} $
$=$ $3 x^{3} + y^{2} z + y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 385 x^{6} + 27 x^{5} z - 285 x^{4} z^{2} - 48 x^{3} y^{3} - 80 x^{3} z^{3} - 72 x^{2} y^{3} z + \cdots + z^{6} $
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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.

Canonical model
$(0:1/3:0:1)$, $(0:-1/5:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle y+\frac{1}{9}w$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle z+\frac{4}{9}w$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^3\cdot5^2}\cdot\frac{230291100000yz^{10}w-259659448200000yz^{8}w^{3}+1427759167104000yz^{6}w^{5}+194177755699200yz^{4}w^{7}+8557798809600yz^{2}w^{9}+124727902208yw^{11}+553584375z^{12}-6273956250000z^{10}w^{2}-464042318310000z^{8}w^{4}+260152515417600z^{6}w^{6}+37614580727040z^{4}w^{8}+1694173360128z^{2}w^{10}+25090699264w^{12}}{z^{2}(164025000yz^{8}w-506655000yz^{6}w^{3}+449695800yz^{4}w^{5}-157807080yz^{2}w^{7}+19413152yw^{9}-12301875z^{10}+158557500z^{8}w^{2}-246584250z^{6}w^{4}+148703580z^{4}w^{6}-39727083z^{2}w^{8}+3945616w^{10})}$

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
12.36.2.t.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$
24.24.0.bl.1 $24$ $3$ $3$ $0$ $0$ full Jacobian
24.36.2.cj.1 $24$ $2$ $2$ $2$ $0$ $1^{2}$
24.36.2.cl.1 $24$ $2$ $2$ $2$ $0$ $1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.144.7.vh.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.vj.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.vp.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.vr.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.yj.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.yl.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.yr.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.yt.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.8.go.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.go.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gp.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gp.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gq.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gq.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gr.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.gr.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dk.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dk.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dl.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dl.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dm.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dm.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dn.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.dn.2 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.9.cq.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.cs.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.gy.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.gz.1 $48$ $2$ $2$ $9$ $0$ $1^{5}$
48.144.9.ka.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.kb.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.lw.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.ly.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
72.216.16.fd.1 $72$ $3$ $3$ $16$ $?$ not computed
120.144.7.dga.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dgc.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dgi.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dgk.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dmy.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dna.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dng.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.dni.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.8.qs.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qs.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qt.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qt.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qu.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qu.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qv.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qv.2 $120$ $2$ $2$ $8$ $?$ not computed
168.144.7.cvd.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cvf.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cvl.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.cvn.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.daf.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.dah.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.dan.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.dap.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.8.qk.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qk.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ql.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ql.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qm.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qm.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qn.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.qn.2 $168$ $2$ $2$ $8$ $?$ not computed
240.144.8.ei.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ei.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ej.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ej.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ek.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.ek.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.el.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.el.2 $240$ $2$ $2$ $8$ $?$ not computed
240.144.9.ie.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.if.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.jk.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.jl.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.mq.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.mr.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.om.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.on.1 $240$ $2$ $2$ $9$ $?$ not computed
264.144.7.cvd.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cvf.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cvl.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.cvn.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.daf.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.dah.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.dan.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.dap.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.8.qm.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qm.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qn.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qn.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qo.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qo.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qp.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.qp.2 $264$ $2$ $2$ $8$ $?$ not computed
312.144.7.daj.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dal.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dar.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dat.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dfl.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dfn.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dft.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.dfv.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.8.qs.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qs.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qt.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qt.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qu.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qu.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qv.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qv.2 $312$ $2$ $2$ $8$ $?$ not computed