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Properties

Label 24.48.1.iu.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: $144$
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: 24.48.1.503

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&3\\0&17\end{bmatrix}$, $\begin{bmatrix}5&12\\8&17\end{bmatrix}$, $\begin{bmatrix}5&18\\4&13\end{bmatrix}$, $\begin{bmatrix}7&9\\4&19\end{bmatrix}$, $\begin{bmatrix}13&21\\4&19\end{bmatrix}$, $\begin{bmatrix}23&18\\0&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.iu.1.1, 24.96.1-24.iu.1.2, 24.96.1-24.iu.1.3, 24.96.1-24.iu.1.4, 24.96.1-24.iu.1.5, 24.96.1-24.iu.1.6, 24.96.1-24.iu.1.7, 24.96.1-24.iu.1.8, 24.96.1-24.iu.1.9, 24.96.1-24.iu.1.10, 24.96.1-24.iu.1.11, 24.96.1-24.iu.1.12, 24.96.1-24.iu.1.13, 24.96.1-24.iu.1.14, 24.96.1-24.iu.1.15, 24.96.1-24.iu.1.16, 24.96.1-24.iu.1.17, 24.96.1-24.iu.1.18, 24.96.1-24.iu.1.19, 24.96.1-24.iu.1.20, 24.96.1-24.iu.1.21, 24.96.1-24.iu.1.22, 24.96.1-24.iu.1.23, 24.96.1-24.iu.1.24, 24.96.1-24.iu.1.25, 24.96.1-24.iu.1.26, 24.96.1-24.iu.1.27, 24.96.1-24.iu.1.28, 24.96.1-24.iu.1.29, 24.96.1-24.iu.1.30, 24.96.1-24.iu.1.31, 24.96.1-24.iu.1.32, 120.96.1-24.iu.1.1, 120.96.1-24.iu.1.2, 120.96.1-24.iu.1.3, 120.96.1-24.iu.1.4, 120.96.1-24.iu.1.5, 120.96.1-24.iu.1.6, 120.96.1-24.iu.1.7, 120.96.1-24.iu.1.8, 120.96.1-24.iu.1.9, 120.96.1-24.iu.1.10, 120.96.1-24.iu.1.11, 120.96.1-24.iu.1.12, 120.96.1-24.iu.1.13, 120.96.1-24.iu.1.14, 120.96.1-24.iu.1.15, 120.96.1-24.iu.1.16, 120.96.1-24.iu.1.17, 120.96.1-24.iu.1.18, 120.96.1-24.iu.1.19, 120.96.1-24.iu.1.20, 120.96.1-24.iu.1.21, 120.96.1-24.iu.1.22, 120.96.1-24.iu.1.23, 120.96.1-24.iu.1.24, 120.96.1-24.iu.1.25, 120.96.1-24.iu.1.26, 120.96.1-24.iu.1.27, 120.96.1-24.iu.1.28, 120.96.1-24.iu.1.29, 120.96.1-24.iu.1.30, 120.96.1-24.iu.1.31, 120.96.1-24.iu.1.32, 168.96.1-24.iu.1.1, 168.96.1-24.iu.1.2, 168.96.1-24.iu.1.3, 168.96.1-24.iu.1.4, 168.96.1-24.iu.1.5, 168.96.1-24.iu.1.6, 168.96.1-24.iu.1.7, 168.96.1-24.iu.1.8, 168.96.1-24.iu.1.9, 168.96.1-24.iu.1.10, 168.96.1-24.iu.1.11, 168.96.1-24.iu.1.12, 168.96.1-24.iu.1.13, 168.96.1-24.iu.1.14, 168.96.1-24.iu.1.15, 168.96.1-24.iu.1.16, 168.96.1-24.iu.1.17, 168.96.1-24.iu.1.18, 168.96.1-24.iu.1.19, 168.96.1-24.iu.1.20, 168.96.1-24.iu.1.21, 168.96.1-24.iu.1.22, 168.96.1-24.iu.1.23, 168.96.1-24.iu.1.24, 168.96.1-24.iu.1.25, 168.96.1-24.iu.1.26, 168.96.1-24.iu.1.27, 168.96.1-24.iu.1.28, 168.96.1-24.iu.1.29, 168.96.1-24.iu.1.30, 168.96.1-24.iu.1.31, 168.96.1-24.iu.1.32, 264.96.1-24.iu.1.1, 264.96.1-24.iu.1.2, 264.96.1-24.iu.1.3, 264.96.1-24.iu.1.4, 264.96.1-24.iu.1.5, 264.96.1-24.iu.1.6, 264.96.1-24.iu.1.7, 264.96.1-24.iu.1.8, 264.96.1-24.iu.1.9, 264.96.1-24.iu.1.10, 264.96.1-24.iu.1.11, 264.96.1-24.iu.1.12, 264.96.1-24.iu.1.13, 264.96.1-24.iu.1.14, 264.96.1-24.iu.1.15, 264.96.1-24.iu.1.16, 264.96.1-24.iu.1.17, 264.96.1-24.iu.1.18, 264.96.1-24.iu.1.19, 264.96.1-24.iu.1.20, 264.96.1-24.iu.1.21, 264.96.1-24.iu.1.22, 264.96.1-24.iu.1.23, 264.96.1-24.iu.1.24, 264.96.1-24.iu.1.25, 264.96.1-24.iu.1.26, 264.96.1-24.iu.1.27, 264.96.1-24.iu.1.28, 264.96.1-24.iu.1.29, 264.96.1-24.iu.1.30, 264.96.1-24.iu.1.31, 264.96.1-24.iu.1.32, 312.96.1-24.iu.1.1, 312.96.1-24.iu.1.2, 312.96.1-24.iu.1.3, 312.96.1-24.iu.1.4, 312.96.1-24.iu.1.5, 312.96.1-24.iu.1.6, 312.96.1-24.iu.1.7, 312.96.1-24.iu.1.8, 312.96.1-24.iu.1.9, 312.96.1-24.iu.1.10, 312.96.1-24.iu.1.11, 312.96.1-24.iu.1.12, 312.96.1-24.iu.1.13, 312.96.1-24.iu.1.14, 312.96.1-24.iu.1.15, 312.96.1-24.iu.1.16, 312.96.1-24.iu.1.17, 312.96.1-24.iu.1.18, 312.96.1-24.iu.1.19, 312.96.1-24.iu.1.20, 312.96.1-24.iu.1.21, 312.96.1-24.iu.1.22, 312.96.1-24.iu.1.23, 312.96.1-24.iu.1.24, 312.96.1-24.iu.1.25, 312.96.1-24.iu.1.26, 312.96.1-24.iu.1.27, 312.96.1-24.iu.1.28, 312.96.1-24.iu.1.29, 312.96.1-24.iu.1.30, 312.96.1-24.iu.1.31, 312.96.1-24.iu.1.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{4}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 144.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 39x + 70 $
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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
$(0:1:0)$, $(-7:0:1)$, $(2:0:1)$, $(5:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^4}\cdot\frac{48x^{2}y^{14}-250290x^{2}y^{12}z^{2}+704021544x^{2}y^{10}z^{4}-1036544079285x^{2}y^{8}z^{6}+792049398959880x^{2}y^{6}z^{8}-328014454394251449x^{2}y^{4}z^{10}+61839042257985762780x^{2}y^{2}z^{12}-4145763953672617760757x^{2}z^{14}-1092xy^{14}z+4500360xy^{12}z^{3}-9035067807xy^{10}z^{5}+11068238211714xy^{8}z^{7}-7707764288002176xy^{6}z^{9}+2801618589805099920xy^{4}z^{11}-472047366323568417561xy^{2}z^{13}+29056820286475034636310xz^{15}-y^{16}+15600y^{14}z^{2}-58606740y^{12}z^{4}+92100693600y^{10}z^{6}-80931580839972y^{8}z^{8}+42434572546725408y^{6}z^{10}-10661845166918803242y^{4}z^{12}+1158729840347955398568y^{2}z^{14}-41640003747391110588801z^{16}}{z^{2}y^{2}(x^{2}y^{10}-9828x^{2}y^{8}z^{2}+6271587x^{2}y^{6}z^{4}-873137880x^{2}y^{4}z^{6}-68024448x^{2}y^{2}z^{8}-918330048x^{2}z^{10}-40xy^{10}z+112779xy^{8}z^{3}-52069554xy^{6}z^{5}+6093069480xy^{4}z^{7}-365631408xy^{2}z^{9}-4591650240xz^{11}+742y^{10}z^{2}-784512y^{8}z^{4}+175578192y^{6}z^{6}-8713427904y^{4}z^{8}+697250592y^{2}z^{10}+12856620672z^{12})}$

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.ba.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.dl.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dl.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dl.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dl.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dn.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dn.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dn.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.dn.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.ch.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cp.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.em.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.eo.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.ez.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.fa.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ga.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gd.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.gs.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gs.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gs.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gs.4 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gu.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gu.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gu.3 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.gu.4 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.er.1 $24$ $3$ $3$ $5$ $0$ $1^{4}$
72.144.5.bo.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.dc.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.dk.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.tb.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tb.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tb.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.tb.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.td.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.td.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.td.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.td.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.os.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ou.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.py.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sy.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sy.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sy.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sy.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ta.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ta.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ta.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ta.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.brm.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkgy.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.sz.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sz.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sz.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sz.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tb.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tb.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tb.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.tb.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.me.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mo.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nk.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ns.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nu.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qk.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qk.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qk.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qk.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qm.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qm.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qm.4 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.sz.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sz.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sz.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sz.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tb.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tb.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tb.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.tb.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.me.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mo.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nk.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ns.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nu.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qk.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qk.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qk.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qk.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qm.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qm.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qm.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.tb.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tb.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tb.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.tb.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.td.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.td.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.td.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.td.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.os.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ou.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pa.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.py.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qa.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qg.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qi.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sy.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sy.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sy.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sy.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ta.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ta.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ta.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ta.4 $312$ $2$ $2$ $3$ $?$ not computed