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Properties

Label 24.96.1.dr.2
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: $48$
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 $1^{2}\cdot2^{5}\cdot4$
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: 24.96.1.1596

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&0\\16&13\end{bmatrix}$, $\begin{bmatrix}5&9\\12&5\end{bmatrix}$, $\begin{bmatrix}7&0\\16&11\end{bmatrix}$, $\begin{bmatrix}11&12\\0&17\end{bmatrix}$, $\begin{bmatrix}11&21\\20&13\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1033088
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dr.2.1, 24.192.1-24.dr.2.2, 24.192.1-24.dr.2.3, 24.192.1-24.dr.2.4, 24.192.1-24.dr.2.5, 24.192.1-24.dr.2.6, 24.192.1-24.dr.2.7, 24.192.1-24.dr.2.8, 24.192.1-24.dr.2.9, 24.192.1-24.dr.2.10, 24.192.1-24.dr.2.11, 24.192.1-24.dr.2.12, 24.192.1-24.dr.2.13, 24.192.1-24.dr.2.14, 24.192.1-24.dr.2.15, 24.192.1-24.dr.2.16, 48.192.1-24.dr.2.1, 48.192.1-24.dr.2.2, 48.192.1-24.dr.2.3, 48.192.1-24.dr.2.4, 48.192.1-24.dr.2.5, 48.192.1-24.dr.2.6, 48.192.1-24.dr.2.7, 48.192.1-24.dr.2.8, 120.192.1-24.dr.2.1, 120.192.1-24.dr.2.2, 120.192.1-24.dr.2.3, 120.192.1-24.dr.2.4, 120.192.1-24.dr.2.5, 120.192.1-24.dr.2.6, 120.192.1-24.dr.2.7, 120.192.1-24.dr.2.8, 120.192.1-24.dr.2.9, 120.192.1-24.dr.2.10, 120.192.1-24.dr.2.11, 120.192.1-24.dr.2.12, 120.192.1-24.dr.2.13, 120.192.1-24.dr.2.14, 120.192.1-24.dr.2.15, 120.192.1-24.dr.2.16, 168.192.1-24.dr.2.1, 168.192.1-24.dr.2.2, 168.192.1-24.dr.2.3, 168.192.1-24.dr.2.4, 168.192.1-24.dr.2.5, 168.192.1-24.dr.2.6, 168.192.1-24.dr.2.7, 168.192.1-24.dr.2.8, 168.192.1-24.dr.2.9, 168.192.1-24.dr.2.10, 168.192.1-24.dr.2.11, 168.192.1-24.dr.2.12, 168.192.1-24.dr.2.13, 168.192.1-24.dr.2.14, 168.192.1-24.dr.2.15, 168.192.1-24.dr.2.16, 240.192.1-24.dr.2.1, 240.192.1-24.dr.2.2, 240.192.1-24.dr.2.3, 240.192.1-24.dr.2.4, 240.192.1-24.dr.2.5, 240.192.1-24.dr.2.6, 240.192.1-24.dr.2.7, 240.192.1-24.dr.2.8, 264.192.1-24.dr.2.1, 264.192.1-24.dr.2.2, 264.192.1-24.dr.2.3, 264.192.1-24.dr.2.4, 264.192.1-24.dr.2.5, 264.192.1-24.dr.2.6, 264.192.1-24.dr.2.7, 264.192.1-24.dr.2.8, 264.192.1-24.dr.2.9, 264.192.1-24.dr.2.10, 264.192.1-24.dr.2.11, 264.192.1-24.dr.2.12, 264.192.1-24.dr.2.13, 264.192.1-24.dr.2.14, 264.192.1-24.dr.2.15, 264.192.1-24.dr.2.16, 312.192.1-24.dr.2.1, 312.192.1-24.dr.2.2, 312.192.1-24.dr.2.3, 312.192.1-24.dr.2.4, 312.192.1-24.dr.2.5, 312.192.1-24.dr.2.6, 312.192.1-24.dr.2.7, 312.192.1-24.dr.2.8, 312.192.1-24.dr.2.9, 312.192.1-24.dr.2.10, 312.192.1-24.dr.2.11, 312.192.1-24.dr.2.12, 312.192.1-24.dr.2.13, 312.192.1-24.dr.2.14, 312.192.1-24.dr.2.15, 312.192.1-24.dr.2.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{4}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 48.2.a.a

Models

Weierstrass model Weierstrass model

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

This modular curve has 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
$(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{8x^{2}y^{30}-72x^{2}y^{28}z^{2}-3000x^{2}y^{26}z^{4}-11896x^{2}y^{24}z^{6}-11064x^{2}y^{22}z^{8}+45640x^{2}y^{20}z^{10}+188328x^{2}y^{18}z^{12}+370008x^{2}y^{16}z^{14}+447192x^{2}y^{14}z^{16}+316936x^{2}y^{12}z^{18}+78552x^{2}y^{10}z^{20}-85512x^{2}y^{8}z^{22}-104104x^{2}y^{6}z^{24}-48648x^{2}y^{4}z^{26}-10184x^{2}y^{2}z^{28}-728x^{2}z^{30}+8xy^{30}z+648xy^{28}z^{3}-168xy^{26}z^{5}-18760xy^{24}z^{7}-72792xy^{22}z^{9}-144568xy^{20}z^{11}-127368xy^{18}z^{13}+72504xy^{16}z^{15}+389016xy^{14}z^{17}+599896xy^{12}z^{19}+543240xy^{10}z^{21}+301800xy^{8}z^{23}+84152xy^{6}z^{25}+216xy^{4}z^{27}-5080xy^{2}z^{29}-728xz^{31}+y^{32}-96y^{30}z^{2}+312y^{28}z^{4}+7696y^{26}z^{6}+28204y^{24}z^{8}+48256y^{22}z^{10}+23256y^{20}z^{12}-103728y^{18}z^{14}-284058y^{16}z^{16}-389728y^{14}z^{18}-334680y^{12}z^{20}-178896y^{10}z^{22}-44564y^{8}z^{24}+5440y^{6}z^{26}+5320y^{4}z^{28}+752y^{2}z^{30}+z^{32}}{z^{8}y^{8}(y^{2}+z^{2})^{3}(30x^{2}y^{6}z^{2}+82x^{2}y^{4}z^{4}+36x^{2}y^{2}z^{6}-12xy^{8}z-36xy^{6}z^{3}+37xy^{4}z^{5}+54xy^{2}z^{7}+9xz^{9}+y^{10}+9y^{8}z^{2}-28y^{6}z^{4}-45y^{4}z^{6}-9y^{2}z^{8})}$

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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bt.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.iw.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.df.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dl.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.en.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.fj.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fp.4 $24$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
24.192.5.fr.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fx.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.ba.2 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
48.192.5.ku.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.kx.4 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.lc.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.lf.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.9.bbp.1 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.192.9.bbs.1 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.bgf.2 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
48.192.9.bgi.2 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
72.288.9.bb.1 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.fc.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fs.2 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.bat.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bav.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdf.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdh.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdx.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.bat.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bav.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbj.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbl.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdf.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdh.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdv.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdx.2 $168$ $2$ $2$ $5$ $?$ not computed
240.192.5.cko.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ckp.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cle.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.clf.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.frj.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.frk.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsp.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.fsq.2 $240$ $2$ $2$ $9$ $?$ not computed
264.192.5.bat.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bav.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbj.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbl.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdf.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdh.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdv.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdx.4 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.bat.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bav.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbj.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbl.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdf.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdh.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdv.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdx.2 $312$ $2$ $2$ $5$ $?$ not computed