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Properties

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

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 $4$ are rational) Cusp widths $6^{4}\cdot24^{2}$ Cusp orbits $1^{4}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24D4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.72.4.14

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&13\\16&11\end{bmatrix}$, $\begin{bmatrix}1&17\\16&7\end{bmatrix}$, $\begin{bmatrix}1&20\\16&13\end{bmatrix}$, $\begin{bmatrix}11&5\\16&17\end{bmatrix}$, $\begin{bmatrix}13&6\\0&13\end{bmatrix}$, $\begin{bmatrix}17&17\\16&11\end{bmatrix}$, $\begin{bmatrix}17&18\\0&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.144.4-24.dh.1.1, 24.144.4-24.dh.1.2, 24.144.4-24.dh.1.3, 24.144.4-24.dh.1.4, 24.144.4-24.dh.1.5, 24.144.4-24.dh.1.6, 24.144.4-24.dh.1.7, 24.144.4-24.dh.1.8, 24.144.4-24.dh.1.9, 24.144.4-24.dh.1.10, 24.144.4-24.dh.1.11, 24.144.4-24.dh.1.12, 24.144.4-24.dh.1.13, 24.144.4-24.dh.1.14, 24.144.4-24.dh.1.15, 24.144.4-24.dh.1.16, 24.144.4-24.dh.1.17, 24.144.4-24.dh.1.18, 24.144.4-24.dh.1.19, 24.144.4-24.dh.1.20, 24.144.4-24.dh.1.21, 24.144.4-24.dh.1.22, 24.144.4-24.dh.1.23, 24.144.4-24.dh.1.24, 48.144.4-24.dh.1.1, 48.144.4-24.dh.1.2, 48.144.4-24.dh.1.3, 48.144.4-24.dh.1.4, 48.144.4-24.dh.1.5, 48.144.4-24.dh.1.6, 48.144.4-24.dh.1.7, 48.144.4-24.dh.1.8, 48.144.4-24.dh.1.9, 48.144.4-24.dh.1.10, 48.144.4-24.dh.1.11, 48.144.4-24.dh.1.12, 48.144.4-24.dh.1.13, 48.144.4-24.dh.1.14, 48.144.4-24.dh.1.15, 48.144.4-24.dh.1.16, 48.144.4-24.dh.1.17, 48.144.4-24.dh.1.18, 48.144.4-24.dh.1.19, 48.144.4-24.dh.1.20, 120.144.4-24.dh.1.1, 120.144.4-24.dh.1.2, 120.144.4-24.dh.1.3, 120.144.4-24.dh.1.4, 120.144.4-24.dh.1.5, 120.144.4-24.dh.1.6, 120.144.4-24.dh.1.7, 120.144.4-24.dh.1.8, 120.144.4-24.dh.1.9, 120.144.4-24.dh.1.10, 120.144.4-24.dh.1.11, 120.144.4-24.dh.1.12, 120.144.4-24.dh.1.13, 120.144.4-24.dh.1.14, 120.144.4-24.dh.1.15, 120.144.4-24.dh.1.16, 120.144.4-24.dh.1.17, 120.144.4-24.dh.1.18, 120.144.4-24.dh.1.19, 120.144.4-24.dh.1.20, 120.144.4-24.dh.1.21, 120.144.4-24.dh.1.22, 120.144.4-24.dh.1.23, 120.144.4-24.dh.1.24, 168.144.4-24.dh.1.1, 168.144.4-24.dh.1.2, 168.144.4-24.dh.1.3, 168.144.4-24.dh.1.4, 168.144.4-24.dh.1.5, 168.144.4-24.dh.1.6, 168.144.4-24.dh.1.7, 168.144.4-24.dh.1.8, 168.144.4-24.dh.1.9, 168.144.4-24.dh.1.10, 168.144.4-24.dh.1.11, 168.144.4-24.dh.1.12, 168.144.4-24.dh.1.13, 168.144.4-24.dh.1.14, 168.144.4-24.dh.1.15, 168.144.4-24.dh.1.16, 168.144.4-24.dh.1.17, 168.144.4-24.dh.1.18, 168.144.4-24.dh.1.19, 168.144.4-24.dh.1.20, 168.144.4-24.dh.1.21, 168.144.4-24.dh.1.22, 168.144.4-24.dh.1.23, 168.144.4-24.dh.1.24, 240.144.4-24.dh.1.1, 240.144.4-24.dh.1.2, 240.144.4-24.dh.1.3, 240.144.4-24.dh.1.4, 240.144.4-24.dh.1.5, 240.144.4-24.dh.1.6, 240.144.4-24.dh.1.7, 240.144.4-24.dh.1.8, 240.144.4-24.dh.1.9, 240.144.4-24.dh.1.10, 240.144.4-24.dh.1.11, 240.144.4-24.dh.1.12, 240.144.4-24.dh.1.13, 240.144.4-24.dh.1.14, 240.144.4-24.dh.1.15, 240.144.4-24.dh.1.16, 240.144.4-24.dh.1.17, 240.144.4-24.dh.1.18, 240.144.4-24.dh.1.19, 240.144.4-24.dh.1.20, 264.144.4-24.dh.1.1, 264.144.4-24.dh.1.2, 264.144.4-24.dh.1.3, 264.144.4-24.dh.1.4, 264.144.4-24.dh.1.5, 264.144.4-24.dh.1.6, 264.144.4-24.dh.1.7, 264.144.4-24.dh.1.8, 264.144.4-24.dh.1.9, 264.144.4-24.dh.1.10, 264.144.4-24.dh.1.11, 264.144.4-24.dh.1.12, 264.144.4-24.dh.1.13, 264.144.4-24.dh.1.14, 264.144.4-24.dh.1.15, 264.144.4-24.dh.1.16, 264.144.4-24.dh.1.17, 264.144.4-24.dh.1.18, 264.144.4-24.dh.1.19, 264.144.4-24.dh.1.20, 264.144.4-24.dh.1.21, 264.144.4-24.dh.1.22, 264.144.4-24.dh.1.23, 264.144.4-24.dh.1.24, 312.144.4-24.dh.1.1, 312.144.4-24.dh.1.2, 312.144.4-24.dh.1.3, 312.144.4-24.dh.1.4, 312.144.4-24.dh.1.5, 312.144.4-24.dh.1.6, 312.144.4-24.dh.1.7, 312.144.4-24.dh.1.8, 312.144.4-24.dh.1.9, 312.144.4-24.dh.1.10, 312.144.4-24.dh.1.11, 312.144.4-24.dh.1.12, 312.144.4-24.dh.1.13, 312.144.4-24.dh.1.14, 312.144.4-24.dh.1.15, 312.144.4-24.dh.1.16, 312.144.4-24.dh.1.17, 312.144.4-24.dh.1.18, 312.144.4-24.dh.1.19, 312.144.4-24.dh.1.20, 312.144.4-24.dh.1.21, 312.144.4-24.dh.1.22, 312.144.4-24.dh.1.23, 312.144.4-24.dh.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 $ $=$ $ 16 y^{2} - z^{2} - w^{2} $
$=$ $x^{3} + y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - x^{4} z + 2 x^{2} y^{3} + z^{5} $
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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.

Canonical model
$(0:-1/4:0:1)$, $(0:1/4:1:0)$, $(0:1/4:0:1)$, $(0:-1/4:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{(z^{2}-4zw+w^{2})^{3}(z^{2}+4zw+w^{2})^{3}}{w^{2}z^{2}(z^{2}+w^{2})^{4}}$

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
$X_{\mathrm{ns}}^+(3)$ $3$ $24$ $24$ $0$ $0$ full Jacobian
8.24.0.q.1 $8$ $3$ $3$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.q.1 $8$ $3$ $3$ $0$ $0$ full Jacobian
12.36.2.p.1 $12$ $2$ $2$ $2$ $0$ $1^{2}$
24.36.1.fp.1 $24$ $2$ $2$ $1$ $0$ $1^{3}$
24.36.2.cj.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.mt.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.mu.1 $24$ $2$ $2$ $7$ $2$ $1^{3}$
24.144.7.nb.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.nc.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.nj.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.nk.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.7.nr.1 $24$ $2$ $2$ $7$ $1$ $1^{3}$
24.144.7.ns.1 $24$ $2$ $2$ $7$ $0$ $1^{3}$
24.144.8.fu.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fu.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fv.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fv.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fw.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fw.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fx.1 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.8.fx.2 $24$ $2$ $2$ $8$ $0$ $2^{2}$
24.144.9.uu.1 $24$ $2$ $2$ $9$ $1$ $1^{5}$
24.144.9.uv.1 $24$ $2$ $2$ $9$ $1$ $1^{5}$
24.144.9.uw.1 $24$ $2$ $2$ $9$ $1$ $1^{5}$
24.144.9.ux.1 $24$ $2$ $2$ $9$ $2$ $1^{5}$
24.144.9.uy.1 $24$ $2$ $2$ $9$ $1$ $1^{5}$
24.144.9.uz.1 $24$ $2$ $2$ $9$ $0$ $1^{5}$
24.144.9.va.1 $24$ $2$ $2$ $9$ $2$ $1^{5}$
24.144.9.vb.1 $24$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.8.bk.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bl.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bm.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bn.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bo.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bp.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.bq.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.8.br.1 $48$ $2$ $2$ $8$ $0$ $2^{2}$
48.144.9.y.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.z.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.ba.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.bb.1 $48$ $2$ $2$ $9$ $0$ $1^{5}$
48.144.9.bc.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.bd.1 $48$ $2$ $2$ $9$ $2$ $1^{5}$
48.144.9.be.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.9.bf.1 $48$ $2$ $2$ $9$ $1$ $1^{5}$
48.144.10.bf.1 $48$ $2$ $2$ $10$ $0$ $2\cdot4$
48.144.10.bg.1 $48$ $2$ $2$ $10$ $0$ $2\cdot4$
48.144.10.bh.1 $48$ $2$ $2$ $10$ $0$ $2\cdot4$
48.144.10.bi.1 $48$ $2$ $2$ $10$ $0$ $2\cdot4$
72.216.16.dh.1 $72$ $3$ $3$ $16$ $?$ not computed
120.144.7.bzm.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.bzn.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.bzu.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.bzv.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.cac.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.cad.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.cak.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.cal.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.8.py.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.py.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.pz.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.pz.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qa.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qa.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qb.1 $120$ $2$ $2$ $8$ $?$ not computed
120.144.8.qb.2 $120$ $2$ $2$ $8$ $?$ not computed
120.144.9.cjc.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cjd.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cje.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cjf.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cjg.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cjh.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cji.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.cjj.1 $120$ $2$ $2$ $9$ $?$ not computed
168.144.7.bvn.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bvo.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bvv.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bvw.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bwd.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bwe.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bwl.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.7.bwm.1 $168$ $2$ $2$ $7$ $?$ not computed
168.144.8.pq.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.pq.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.pr.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.pr.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ps.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.ps.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.pt.1 $168$ $2$ $2$ $8$ $?$ not computed
168.144.8.pt.2 $168$ $2$ $2$ $8$ $?$ not computed
168.144.9.cie.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cif.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cig.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cih.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cii.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cij.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cik.1 $168$ $2$ $2$ $9$ $?$ not computed
168.144.9.cil.1 $168$ $2$ $2$ $9$ $?$ not computed
240.144.8.bs.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bt.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bu.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bv.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bw.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bx.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.by.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.8.bz.1 $240$ $2$ $2$ $8$ $?$ not computed
240.144.9.y.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.z.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.ba.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bb.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bc.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bd.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.be.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.9.bf.1 $240$ $2$ $2$ $9$ $?$ not computed
240.144.10.bk.1 $240$ $2$ $2$ $10$ $?$ not computed
240.144.10.bl.1 $240$ $2$ $2$ $10$ $?$ not computed
240.144.10.bm.1 $240$ $2$ $2$ $10$ $?$ not computed
240.144.10.bn.1 $240$ $2$ $2$ $10$ $?$ not computed
264.144.7.bvn.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bvo.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bvv.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bvw.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bwd.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bwe.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bwl.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.7.bwm.1 $264$ $2$ $2$ $7$ $?$ not computed
264.144.8.pr.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.pr.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.ps.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.ps.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.pt.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.pt.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.pu.1 $264$ $2$ $2$ $8$ $?$ not computed
264.144.8.pu.2 $264$ $2$ $2$ $8$ $?$ not computed
264.144.9.ciq.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.cir.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.cis.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.cit.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.ciu.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.civ.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.ciw.1 $264$ $2$ $2$ $9$ $?$ not computed
264.144.9.cix.1 $264$ $2$ $2$ $9$ $?$ not computed
312.144.7.bvn.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bvo.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bvv.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bvw.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bwd.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bwe.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bwl.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.7.bwm.1 $312$ $2$ $2$ $7$ $?$ not computed
312.144.8.py.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.py.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.pz.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.pz.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qa.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qa.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qb.1 $312$ $2$ $2$ $8$ $?$ not computed
312.144.8.qb.2 $312$ $2$ $2$ $8$ $?$ not computed
312.144.9.cie.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cif.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cig.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cih.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cii.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cij.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cik.1 $312$ $2$ $2$ $9$ $?$ not computed
312.144.9.cil.1 $312$ $2$ $2$ $9$ $?$ not computed