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Properties

Label 24.96.1.dg.4
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $24$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{4}\cdot2^{6}$
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: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1609

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&14\\0&19\end{bmatrix}$, $\begin{bmatrix}5&1\\0&13\end{bmatrix}$, $\begin{bmatrix}13&14\\0&13\end{bmatrix}$, $\begin{bmatrix}17&4\\0&1\end{bmatrix}$, $\begin{bmatrix}19&8\\0&23\end{bmatrix}$, $\begin{bmatrix}19&9\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_{24}:C_2^5$
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dg.4.1, 24.192.1-24.dg.4.2, 24.192.1-24.dg.4.3, 24.192.1-24.dg.4.4, 24.192.1-24.dg.4.5, 24.192.1-24.dg.4.6, 24.192.1-24.dg.4.7, 24.192.1-24.dg.4.8, 24.192.1-24.dg.4.9, 24.192.1-24.dg.4.10, 24.192.1-24.dg.4.11, 24.192.1-24.dg.4.12, 24.192.1-24.dg.4.13, 24.192.1-24.dg.4.14, 24.192.1-24.dg.4.15, 24.192.1-24.dg.4.16, 24.192.1-24.dg.4.17, 24.192.1-24.dg.4.18, 24.192.1-24.dg.4.19, 24.192.1-24.dg.4.20, 24.192.1-24.dg.4.21, 24.192.1-24.dg.4.22, 24.192.1-24.dg.4.23, 24.192.1-24.dg.4.24, 48.192.1-24.dg.4.1, 48.192.1-24.dg.4.2, 48.192.1-24.dg.4.3, 48.192.1-24.dg.4.4, 48.192.1-24.dg.4.5, 48.192.1-24.dg.4.6, 48.192.1-24.dg.4.7, 48.192.1-24.dg.4.8, 48.192.1-24.dg.4.9, 48.192.1-24.dg.4.10, 48.192.1-24.dg.4.11, 48.192.1-24.dg.4.12, 48.192.1-24.dg.4.13, 48.192.1-24.dg.4.14, 48.192.1-24.dg.4.15, 48.192.1-24.dg.4.16, 120.192.1-24.dg.4.1, 120.192.1-24.dg.4.2, 120.192.1-24.dg.4.3, 120.192.1-24.dg.4.4, 120.192.1-24.dg.4.5, 120.192.1-24.dg.4.6, 120.192.1-24.dg.4.7, 120.192.1-24.dg.4.8, 120.192.1-24.dg.4.9, 120.192.1-24.dg.4.10, 120.192.1-24.dg.4.11, 120.192.1-24.dg.4.12, 120.192.1-24.dg.4.13, 120.192.1-24.dg.4.14, 120.192.1-24.dg.4.15, 120.192.1-24.dg.4.16, 120.192.1-24.dg.4.17, 120.192.1-24.dg.4.18, 120.192.1-24.dg.4.19, 120.192.1-24.dg.4.20, 120.192.1-24.dg.4.21, 120.192.1-24.dg.4.22, 120.192.1-24.dg.4.23, 120.192.1-24.dg.4.24, 168.192.1-24.dg.4.1, 168.192.1-24.dg.4.2, 168.192.1-24.dg.4.3, 168.192.1-24.dg.4.4, 168.192.1-24.dg.4.5, 168.192.1-24.dg.4.6, 168.192.1-24.dg.4.7, 168.192.1-24.dg.4.8, 168.192.1-24.dg.4.9, 168.192.1-24.dg.4.10, 168.192.1-24.dg.4.11, 168.192.1-24.dg.4.12, 168.192.1-24.dg.4.13, 168.192.1-24.dg.4.14, 168.192.1-24.dg.4.15, 168.192.1-24.dg.4.16, 168.192.1-24.dg.4.17, 168.192.1-24.dg.4.18, 168.192.1-24.dg.4.19, 168.192.1-24.dg.4.20, 168.192.1-24.dg.4.21, 168.192.1-24.dg.4.22, 168.192.1-24.dg.4.23, 168.192.1-24.dg.4.24, 240.192.1-24.dg.4.1, 240.192.1-24.dg.4.2, 240.192.1-24.dg.4.3, 240.192.1-24.dg.4.4, 240.192.1-24.dg.4.5, 240.192.1-24.dg.4.6, 240.192.1-24.dg.4.7, 240.192.1-24.dg.4.8, 240.192.1-24.dg.4.9, 240.192.1-24.dg.4.10, 240.192.1-24.dg.4.11, 240.192.1-24.dg.4.12, 240.192.1-24.dg.4.13, 240.192.1-24.dg.4.14, 240.192.1-24.dg.4.15, 240.192.1-24.dg.4.16, 264.192.1-24.dg.4.1, 264.192.1-24.dg.4.2, 264.192.1-24.dg.4.3, 264.192.1-24.dg.4.4, 264.192.1-24.dg.4.5, 264.192.1-24.dg.4.6, 264.192.1-24.dg.4.7, 264.192.1-24.dg.4.8, 264.192.1-24.dg.4.9, 264.192.1-24.dg.4.10, 264.192.1-24.dg.4.11, 264.192.1-24.dg.4.12, 264.192.1-24.dg.4.13, 264.192.1-24.dg.4.14, 264.192.1-24.dg.4.15, 264.192.1-24.dg.4.16, 264.192.1-24.dg.4.17, 264.192.1-24.dg.4.18, 264.192.1-24.dg.4.19, 264.192.1-24.dg.4.20, 264.192.1-24.dg.4.21, 264.192.1-24.dg.4.22, 264.192.1-24.dg.4.23, 264.192.1-24.dg.4.24, 312.192.1-24.dg.4.1, 312.192.1-24.dg.4.2, 312.192.1-24.dg.4.3, 312.192.1-24.dg.4.4, 312.192.1-24.dg.4.5, 312.192.1-24.dg.4.6, 312.192.1-24.dg.4.7, 312.192.1-24.dg.4.8, 312.192.1-24.dg.4.9, 312.192.1-24.dg.4.10, 312.192.1-24.dg.4.11, 312.192.1-24.dg.4.12, 312.192.1-24.dg.4.13, 312.192.1-24.dg.4.14, 312.192.1-24.dg.4.15, 312.192.1-24.dg.4.16, 312.192.1-24.dg.4.17, 312.192.1-24.dg.4.18, 312.192.1-24.dg.4.19, 312.192.1-24.dg.4.20, 312.192.1-24.dg.4.21, 312.192.1-24.dg.4.22, 312.192.1-24.dg.4.23, 312.192.1-24.dg.4.24
Cyclic 24-isogeny field degree: $1$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

Weierstrass model Weierstrass model

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

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{728x^{2}y^{30}-497918088x^{2}y^{28}z^{2}+250596916440x^{2}y^{26}z^{4}-3084740789384x^{2}y^{24}z^{6}+9123402542136x^{2}y^{22}z^{8}-11598446336680x^{2}y^{20}z^{10}+6943921340088x^{2}y^{18}z^{12}-1916333708328x^{2}y^{16}z^{14}+575054061192x^{2}y^{14}z^{16}-628466196376x^{2}y^{12}z^{18}+383187297672x^{2}y^{10}z^{20}-57512029848x^{2}y^{8}z^{22}-45437042584x^{2}y^{6}z^{24}+25956978888x^{2}y^{4}z^{26}-5423886104x^{2}y^{2}z^{28}+387420488x^{2}z^{30}-185048xy^{30}z+6948184968xy^{28}z^{3}-971785364952xy^{26}z^{5}+6267441823880xy^{24}z^{7}-15011790940728xy^{22}z^{9}+17557901882792xy^{20}z^{11}-10638497539512xy^{18}z^{13}+2842197177384xy^{16}z^{15}+179928501624xy^{14}z^{17}-119509559144xy^{12}z^{19}-268535389320xy^{10}z^{21}+207894327960xy^{8}z^{23}-54517636712xy^{6}z^{25}+194616xy^{4}z^{27}+2711942680xy^{2}z^{29}-387420488xz^{31}-y^{32}+18210864y^{30}z^{2}-53560420152y^{28}z^{4}+2197285656976y^{26}z^{6}-9543336429724y^{24}z^{8}+17759713520176y^{22}z^{10}-18061492774536y^{20}z^{12}+11084067639312y^{18}z^{14}-4371532504902y^{16}z^{16}+990114466832y^{14}z^{18}+62257555320y^{12}z^{20}-132726234576y^{10}z^{22}+34802892644y^{8}z^{24}+2345011600y^{6}z^{26}-2712138040y^{4}z^{28}+387421232y^{2}z^{30}-z^{32}}{y^{2}(y-z)(y+z)(x^{2}y^{26}-43x^{2}y^{24}z^{2}-1016x^{2}y^{22}z^{4}+173744x^{2}y^{20}z^{6}-5143503x^{2}y^{18}z^{8}+26333773x^{2}y^{16}z^{10}-22939592x^{2}y^{14}z^{12}-135535704x^{2}y^{12}z^{14}+371476691x^{2}y^{10}z^{16}-349155873x^{2}y^{8}z^{18}+129140416x^{2}y^{6}z^{20}-14348888x^{2}y^{4}z^{22}-5x^{2}y^{2}z^{24}-x^{2}z^{26}+16xy^{26}z-2735xy^{24}z^{3}+151604xy^{22}z^{5}-3378670xy^{20}z^{7}+31678868xy^{18}z^{9}-161958769xy^{16}z^{11}+461360488xy^{14}z^{13}-690854468xy^{12}z^{15}+465544024xy^{10}z^{17}-57396225xy^{8}z^{19}-57395868xy^{6}z^{21}+14348882xy^{4}z^{23}+4xy^{2}z^{25}+xz^{27}+86y^{26}z^{2}-10627y^{24}z^{4}+446262y^{22}z^{6}-7932608y^{20}z^{8}+54485480y^{18}z^{10}-183454659y^{16}z^{12}+308168108y^{14}z^{14}-216845336y^{12}z^{16}-1434y^{10}z^{18}+57396231y^{8}z^{20}-14348674y^{6}z^{22}+24y^{4}z^{24}-4y^{2}z^{26}-z^{28})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bt.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
$X_0(24)$ $24$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.192.5.df.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.do.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dr.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.du.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ev.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fa.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fd.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fi.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.gh.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gi.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gj.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gk.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gl.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gm.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.gn.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.go.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.288.9.p.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.5.ke.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kf.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kg.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kh.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.km.4 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kn.3 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ko.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kp.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.kq.3 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.kt.3 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.ky.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.lb.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.9.bbl.2 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.192.9.bbo.2 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.bgb.1 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
48.192.9.bge.1 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
72.288.9.q.3 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.er.3 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fh.4 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.zg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zi.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.zq.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbs.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbu.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bca.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bcc.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beh.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bei.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bej.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bek.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bel.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bem.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ben.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.beo.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.zg.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zi.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zo.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.zq.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbs.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbu.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bca.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bcc.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beh.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bei.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bej.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bek.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bel.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bem.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ben.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.beo.4 $168$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjo.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjp.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjq.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjr.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjw.4 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjx.4 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjy.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cjz.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckk.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckl.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cla.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.clb.3 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.frf.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.frg.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsl.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsm.1 $240$ $2$ $2$ $9$ $?$ not computed
264.192.5.zg.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zi.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zo.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.zq.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbs.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbu.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bca.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bcc.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beh.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bei.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bej.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bek.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bel.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bem.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ben.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.beo.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.zg.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zi.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zo.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.zq.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbs.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbu.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bca.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bcc.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beh.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bei.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bej.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bek.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bel.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bem.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ben.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.beo.4 $312$ $2$ $2$ $5$ $?$ not computed