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Properties

Label 24.48.1.is.1
Level $24$
Index $48$
Genus $1$
Analytic rank $1$
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: $1$
$\Q$-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: 24.48.1.502

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&6\\0&7\end{bmatrix}$, $\begin{bmatrix}1&6\\20&23\end{bmatrix}$, $\begin{bmatrix}7&0\\12&13\end{bmatrix}$, $\begin{bmatrix}11&9\\8&19\end{bmatrix}$, $\begin{bmatrix}17&0\\16&19\end{bmatrix}$, $\begin{bmatrix}17&21\\16&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.is.1.1, 24.96.1-24.is.1.2, 24.96.1-24.is.1.3, 24.96.1-24.is.1.4, 24.96.1-24.is.1.5, 24.96.1-24.is.1.6, 24.96.1-24.is.1.7, 24.96.1-24.is.1.8, 24.96.1-24.is.1.9, 24.96.1-24.is.1.10, 24.96.1-24.is.1.11, 24.96.1-24.is.1.12, 24.96.1-24.is.1.13, 24.96.1-24.is.1.14, 24.96.1-24.is.1.15, 24.96.1-24.is.1.16, 24.96.1-24.is.1.17, 24.96.1-24.is.1.18, 24.96.1-24.is.1.19, 24.96.1-24.is.1.20, 24.96.1-24.is.1.21, 24.96.1-24.is.1.22, 24.96.1-24.is.1.23, 24.96.1-24.is.1.24, 24.96.1-24.is.1.25, 24.96.1-24.is.1.26, 24.96.1-24.is.1.27, 24.96.1-24.is.1.28, 24.96.1-24.is.1.29, 24.96.1-24.is.1.30, 24.96.1-24.is.1.31, 24.96.1-24.is.1.32, 120.96.1-24.is.1.1, 120.96.1-24.is.1.2, 120.96.1-24.is.1.3, 120.96.1-24.is.1.4, 120.96.1-24.is.1.5, 120.96.1-24.is.1.6, 120.96.1-24.is.1.7, 120.96.1-24.is.1.8, 120.96.1-24.is.1.9, 120.96.1-24.is.1.10, 120.96.1-24.is.1.11, 120.96.1-24.is.1.12, 120.96.1-24.is.1.13, 120.96.1-24.is.1.14, 120.96.1-24.is.1.15, 120.96.1-24.is.1.16, 120.96.1-24.is.1.17, 120.96.1-24.is.1.18, 120.96.1-24.is.1.19, 120.96.1-24.is.1.20, 120.96.1-24.is.1.21, 120.96.1-24.is.1.22, 120.96.1-24.is.1.23, 120.96.1-24.is.1.24, 120.96.1-24.is.1.25, 120.96.1-24.is.1.26, 120.96.1-24.is.1.27, 120.96.1-24.is.1.28, 120.96.1-24.is.1.29, 120.96.1-24.is.1.30, 120.96.1-24.is.1.31, 120.96.1-24.is.1.32, 168.96.1-24.is.1.1, 168.96.1-24.is.1.2, 168.96.1-24.is.1.3, 168.96.1-24.is.1.4, 168.96.1-24.is.1.5, 168.96.1-24.is.1.6, 168.96.1-24.is.1.7, 168.96.1-24.is.1.8, 168.96.1-24.is.1.9, 168.96.1-24.is.1.10, 168.96.1-24.is.1.11, 168.96.1-24.is.1.12, 168.96.1-24.is.1.13, 168.96.1-24.is.1.14, 168.96.1-24.is.1.15, 168.96.1-24.is.1.16, 168.96.1-24.is.1.17, 168.96.1-24.is.1.18, 168.96.1-24.is.1.19, 168.96.1-24.is.1.20, 168.96.1-24.is.1.21, 168.96.1-24.is.1.22, 168.96.1-24.is.1.23, 168.96.1-24.is.1.24, 168.96.1-24.is.1.25, 168.96.1-24.is.1.26, 168.96.1-24.is.1.27, 168.96.1-24.is.1.28, 168.96.1-24.is.1.29, 168.96.1-24.is.1.30, 168.96.1-24.is.1.31, 168.96.1-24.is.1.32, 264.96.1-24.is.1.1, 264.96.1-24.is.1.2, 264.96.1-24.is.1.3, 264.96.1-24.is.1.4, 264.96.1-24.is.1.5, 264.96.1-24.is.1.6, 264.96.1-24.is.1.7, 264.96.1-24.is.1.8, 264.96.1-24.is.1.9, 264.96.1-24.is.1.10, 264.96.1-24.is.1.11, 264.96.1-24.is.1.12, 264.96.1-24.is.1.13, 264.96.1-24.is.1.14, 264.96.1-24.is.1.15, 264.96.1-24.is.1.16, 264.96.1-24.is.1.17, 264.96.1-24.is.1.18, 264.96.1-24.is.1.19, 264.96.1-24.is.1.20, 264.96.1-24.is.1.21, 264.96.1-24.is.1.22, 264.96.1-24.is.1.23, 264.96.1-24.is.1.24, 264.96.1-24.is.1.25, 264.96.1-24.is.1.26, 264.96.1-24.is.1.27, 264.96.1-24.is.1.28, 264.96.1-24.is.1.29, 264.96.1-24.is.1.30, 264.96.1-24.is.1.31, 264.96.1-24.is.1.32, 312.96.1-24.is.1.1, 312.96.1-24.is.1.2, 312.96.1-24.is.1.3, 312.96.1-24.is.1.4, 312.96.1-24.is.1.5, 312.96.1-24.is.1.6, 312.96.1-24.is.1.7, 312.96.1-24.is.1.8, 312.96.1-24.is.1.9, 312.96.1-24.is.1.10, 312.96.1-24.is.1.11, 312.96.1-24.is.1.12, 312.96.1-24.is.1.13, 312.96.1-24.is.1.14, 312.96.1-24.is.1.15, 312.96.1-24.is.1.16, 312.96.1-24.is.1.17, 312.96.1-24.is.1.18, 312.96.1-24.is.1.19, 312.96.1-24.is.1.20, 312.96.1-24.is.1.21, 312.96.1-24.is.1.22, 312.96.1-24.is.1.23, 312.96.1-24.is.1.24, 312.96.1-24.is.1.25, 312.96.1-24.is.1.26, 312.96.1-24.is.1.27, 312.96.1-24.is.1.28, 312.96.1-24.is.1.29, 312.96.1-24.is.1.30, 312.96.1-24.is.1.31, 312.96.1-24.is.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.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 156x + 560 $
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Rational points

This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.

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.y.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.dh.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dh.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dh.3 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dh.4 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dj.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dj.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dj.3 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.dj.4 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.3.cg.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.cp.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ds.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.du.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.fc.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ff.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.ft.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.fu.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.go.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.go.2 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.go.3 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.go.4 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.gq.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.gq.2 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.gq.3 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.gq.4 $24$ $2$ $2$ $3$ $1$ $2$
24.144.5.ev.1 $24$ $3$ $3$ $5$ $2$ $1^{4}$
72.144.5.bm.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.da.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.di.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.sh.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sh.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sh.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sh.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sj.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sj.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.sj.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ok.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.om.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.oo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.oq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ps.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pw.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.se.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.se.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.se.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.se.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sg.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sg.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.sg.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.brc.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkgg.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.sf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sf.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sf.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sf.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sh.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sh.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sh.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.sh.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.lw.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ly.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ma.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.mc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.nc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ne.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ng.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ni.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pq.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pq.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pq.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ps.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ps.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ps.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ps.4 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.sf.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sf.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sf.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sf.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sh.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sh.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sh.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.sh.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.lw.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ly.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ma.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.mc.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.nc.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ne.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ng.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ni.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pq.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pq.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pq.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ps.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ps.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ps.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ps.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.sh.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sh.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sh.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sh.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sj.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sj.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.sj.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ok.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.om.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.oo.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.oq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ps.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pu.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.pw.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.se.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.se.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.se.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.se.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sg.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sg.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sg.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.sg.4 $312$ $2$ $2$ $3$ $?$ not computed