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Properties

Label 24.96.1.ds.1
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
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 $2$ are rational) Cusp widths $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24J1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.2233

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&15\\8&13\end{bmatrix}$, $\begin{bmatrix}1&21\\20&5\end{bmatrix}$, $\begin{bmatrix}13&12\\4&19\end{bmatrix}$, $\begin{bmatrix}17&6\\4&7\end{bmatrix}$, $\begin{bmatrix}19&3\\20&5\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035859
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.ds.1.1, 24.192.1-24.ds.1.2, 24.192.1-24.ds.1.3, 24.192.1-24.ds.1.4, 24.192.1-24.ds.1.5, 24.192.1-24.ds.1.6, 24.192.1-24.ds.1.7, 24.192.1-24.ds.1.8, 24.192.1-24.ds.1.9, 24.192.1-24.ds.1.10, 24.192.1-24.ds.1.11, 24.192.1-24.ds.1.12, 24.192.1-24.ds.1.13, 24.192.1-24.ds.1.14, 24.192.1-24.ds.1.15, 24.192.1-24.ds.1.16, 120.192.1-24.ds.1.1, 120.192.1-24.ds.1.2, 120.192.1-24.ds.1.3, 120.192.1-24.ds.1.4, 120.192.1-24.ds.1.5, 120.192.1-24.ds.1.6, 120.192.1-24.ds.1.7, 120.192.1-24.ds.1.8, 120.192.1-24.ds.1.9, 120.192.1-24.ds.1.10, 120.192.1-24.ds.1.11, 120.192.1-24.ds.1.12, 120.192.1-24.ds.1.13, 120.192.1-24.ds.1.14, 120.192.1-24.ds.1.15, 120.192.1-24.ds.1.16, 168.192.1-24.ds.1.1, 168.192.1-24.ds.1.2, 168.192.1-24.ds.1.3, 168.192.1-24.ds.1.4, 168.192.1-24.ds.1.5, 168.192.1-24.ds.1.6, 168.192.1-24.ds.1.7, 168.192.1-24.ds.1.8, 168.192.1-24.ds.1.9, 168.192.1-24.ds.1.10, 168.192.1-24.ds.1.11, 168.192.1-24.ds.1.12, 168.192.1-24.ds.1.13, 168.192.1-24.ds.1.14, 168.192.1-24.ds.1.15, 168.192.1-24.ds.1.16, 264.192.1-24.ds.1.1, 264.192.1-24.ds.1.2, 264.192.1-24.ds.1.3, 264.192.1-24.ds.1.4, 264.192.1-24.ds.1.5, 264.192.1-24.ds.1.6, 264.192.1-24.ds.1.7, 264.192.1-24.ds.1.8, 264.192.1-24.ds.1.9, 264.192.1-24.ds.1.10, 264.192.1-24.ds.1.11, 264.192.1-24.ds.1.12, 264.192.1-24.ds.1.13, 264.192.1-24.ds.1.14, 264.192.1-24.ds.1.15, 264.192.1-24.ds.1.16, 312.192.1-24.ds.1.1, 312.192.1-24.ds.1.2, 312.192.1-24.ds.1.3, 312.192.1-24.ds.1.4, 312.192.1-24.ds.1.5, 312.192.1-24.ds.1.6, 312.192.1-24.ds.1.7, 312.192.1-24.ds.1.8, 312.192.1-24.ds.1.9, 312.192.1-24.ds.1.10, 312.192.1-24.ds.1.11, 312.192.1-24.ds.1.12, 312.192.1-24.ds.1.13, 312.192.1-24.ds.1.14, 312.192.1-24.ds.1.15, 312.192.1-24.ds.1.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} + 3x + 3 $
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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.

Weierstrass model
$(0:1:0)$, $(-1:0:1)$

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{1}{2^3}\cdot\frac{16x^{2}y^{30}-67968x^{2}y^{28}z^{2}+96568320x^{2}y^{26}z^{4}-53181833216x^{2}y^{24}z^{6}+12262668828672x^{2}y^{22}z^{8}-589865825075200x^{2}y^{20}z^{10}-82154843019411456x^{2}y^{18}z^{12}+877604901683724288x^{2}y^{16}z^{14}-2815046308023238656x^{2}y^{14}z^{16}-1731311060150910976x^{2}y^{12}z^{18}+32445388620023463936x^{2}y^{10}z^{20}-79416707963443740672x^{2}y^{8}z^{22}+73703790543615557632x^{2}y^{6}z^{24}-14759043646800003072x^{2}y^{4}z^{26}-89579411338166272x^{2}y^{2}z^{28}+51228445761339392x^{2}z^{30}-184320xy^{28}z^{3}+567902208xy^{26}z^{5}-484163715072xy^{24}z^{7}+100643553411072xy^{22}z^{9}-8195070918393856xy^{20}z^{11}-115590737097129984xy^{18}z^{13}+2405859487433883648xy^{16}z^{15}-15080689040313286656xy^{14}z^{17}+45600977918455971840xy^{12}z^{19}-65584203386478133248xy^{10}z^{21}+20284379847444135936xy^{8}z^{23}+48137076460870434816xy^{6}z^{25}-34519246718864719872xy^{4}z^{27}+2988138352760324096xy^{2}z^{29}-y^{32}+4976y^{30}z^{2}-8246400y^{28}z^{4}+5157002240y^{26}z^{6}-962136596480y^{24}z^{8}+278644659519488y^{22}z^{10}-36283769801932800y^{20}z^{12}+74798984903786496y^{18}z^{14}+2656867678093836288y^{16}z^{16}-22153717280254459904y^{14}z^{18}+78596841770126934016y^{12}z^{20}-144062141776596566016y^{10}z^{22}+125585164105240543232y^{8}z^{24}-24409104260557438976y^{6}z^{26}-20553082897086545920y^{4}z^{28}+3104176411909292032y^{2}z^{30}-51509920738050048z^{32}}{z^{2}y^{2}(y^{2}-8z^{2})^{3}(12x^{2}y^{20}+928x^{2}y^{18}z^{2}-377856x^{2}y^{16}z^{4}-20865024x^{2}y^{14}z^{6}+33718272x^{2}y^{12}z^{8}+6682312704x^{2}y^{10}z^{10}+18635292672x^{2}y^{8}z^{12}-208540794880x^{2}y^{6}z^{14}-218305134592x^{2}y^{4}z^{16}+913217421312x^{2}y^{2}z^{18}+36xy^{20}z+24448xy^{18}z^{3}+1257216xy^{16}z^{5}-38928384xy^{14}z^{7}-1476231168xy^{12}z^{9}+934281216xy^{10}z^{11}+119367794688xy^{8}z^{13}+7784628224xy^{6}z^{15}-1320903770112xy^{4}z^{17}+521838526464xy^{2}z^{19}+1043677052928xz^{21}-y^{22}-776y^{20}z^{2}-51168y^{18}z^{4}+3945216y^{16}z^{6}+164806656y^{14}z^{8}-335806464y^{12}z^{10}-22092709888y^{10}z^{12}+26778533888y^{8}z^{14}+416762822656y^{6}z^{16}-808930246656y^{4}z^{18}-652298158080y^{2}z^{20}+1043677052928z^{22})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.3 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.ix.1 $24$ $2$ $2$ $1$ $0$ 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.dd.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.do.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ep.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.es.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fl.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fq.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ft.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fy.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.bb.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.bc.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.fd.2 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ft.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bau.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.baw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbk.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdi.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdw.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdy.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bau.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.baw.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbk.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbm.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdg.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdi.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdw.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdy.4 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.bau.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.baw.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbk.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbm.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdg.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdi.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdw.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdy.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bau.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.baw.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbk.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbm.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdg.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdi.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdw.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdy.2 $312$ $2$ $2$ $5$ $?$ not computed