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Properties

Label 24.48.2.g.2
Level $24$
Index $48$
Genus $2$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $6$

Related objects

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $48$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $2^{2}\cdot6^{2}\cdot8\cdot24$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24F2
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.2.12

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&12\\4&13\end{bmatrix}$, $\begin{bmatrix}11&15\\4&1\end{bmatrix}$, $\begin{bmatrix}13&15\\16&17\end{bmatrix}$, $\begin{bmatrix}19&6\\20&5\end{bmatrix}$, $\begin{bmatrix}23&6\\4&19\end{bmatrix}$, $\begin{bmatrix}23&21\\8&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.2-24.g.2.1, 24.96.2-24.g.2.2, 24.96.2-24.g.2.3, 24.96.2-24.g.2.4, 24.96.2-24.g.2.5, 24.96.2-24.g.2.6, 24.96.2-24.g.2.7, 24.96.2-24.g.2.8, 24.96.2-24.g.2.9, 24.96.2-24.g.2.10, 24.96.2-24.g.2.11, 24.96.2-24.g.2.12, 24.96.2-24.g.2.13, 24.96.2-24.g.2.14, 24.96.2-24.g.2.15, 24.96.2-24.g.2.16, 24.96.2-24.g.2.17, 24.96.2-24.g.2.18, 24.96.2-24.g.2.19, 24.96.2-24.g.2.20, 24.96.2-24.g.2.21, 24.96.2-24.g.2.22, 24.96.2-24.g.2.23, 24.96.2-24.g.2.24, 24.96.2-24.g.2.25, 24.96.2-24.g.2.26, 24.96.2-24.g.2.27, 24.96.2-24.g.2.28, 24.96.2-24.g.2.29, 24.96.2-24.g.2.30, 24.96.2-24.g.2.31, 24.96.2-24.g.2.32, 120.96.2-24.g.2.1, 120.96.2-24.g.2.2, 120.96.2-24.g.2.3, 120.96.2-24.g.2.4, 120.96.2-24.g.2.5, 120.96.2-24.g.2.6, 120.96.2-24.g.2.7, 120.96.2-24.g.2.8, 120.96.2-24.g.2.9, 120.96.2-24.g.2.10, 120.96.2-24.g.2.11, 120.96.2-24.g.2.12, 120.96.2-24.g.2.13, 120.96.2-24.g.2.14, 120.96.2-24.g.2.15, 120.96.2-24.g.2.16, 120.96.2-24.g.2.17, 120.96.2-24.g.2.18, 120.96.2-24.g.2.19, 120.96.2-24.g.2.20, 120.96.2-24.g.2.21, 120.96.2-24.g.2.22, 120.96.2-24.g.2.23, 120.96.2-24.g.2.24, 120.96.2-24.g.2.25, 120.96.2-24.g.2.26, 120.96.2-24.g.2.27, 120.96.2-24.g.2.28, 120.96.2-24.g.2.29, 120.96.2-24.g.2.30, 120.96.2-24.g.2.31, 120.96.2-24.g.2.32, 168.96.2-24.g.2.1, 168.96.2-24.g.2.2, 168.96.2-24.g.2.3, 168.96.2-24.g.2.4, 168.96.2-24.g.2.5, 168.96.2-24.g.2.6, 168.96.2-24.g.2.7, 168.96.2-24.g.2.8, 168.96.2-24.g.2.9, 168.96.2-24.g.2.10, 168.96.2-24.g.2.11, 168.96.2-24.g.2.12, 168.96.2-24.g.2.13, 168.96.2-24.g.2.14, 168.96.2-24.g.2.15, 168.96.2-24.g.2.16, 168.96.2-24.g.2.17, 168.96.2-24.g.2.18, 168.96.2-24.g.2.19, 168.96.2-24.g.2.20, 168.96.2-24.g.2.21, 168.96.2-24.g.2.22, 168.96.2-24.g.2.23, 168.96.2-24.g.2.24, 168.96.2-24.g.2.25, 168.96.2-24.g.2.26, 168.96.2-24.g.2.27, 168.96.2-24.g.2.28, 168.96.2-24.g.2.29, 168.96.2-24.g.2.30, 168.96.2-24.g.2.31, 168.96.2-24.g.2.32, 264.96.2-24.g.2.1, 264.96.2-24.g.2.2, 264.96.2-24.g.2.3, 264.96.2-24.g.2.4, 264.96.2-24.g.2.5, 264.96.2-24.g.2.6, 264.96.2-24.g.2.7, 264.96.2-24.g.2.8, 264.96.2-24.g.2.9, 264.96.2-24.g.2.10, 264.96.2-24.g.2.11, 264.96.2-24.g.2.12, 264.96.2-24.g.2.13, 264.96.2-24.g.2.14, 264.96.2-24.g.2.15, 264.96.2-24.g.2.16, 264.96.2-24.g.2.17, 264.96.2-24.g.2.18, 264.96.2-24.g.2.19, 264.96.2-24.g.2.20, 264.96.2-24.g.2.21, 264.96.2-24.g.2.22, 264.96.2-24.g.2.23, 264.96.2-24.g.2.24, 264.96.2-24.g.2.25, 264.96.2-24.g.2.26, 264.96.2-24.g.2.27, 264.96.2-24.g.2.28, 264.96.2-24.g.2.29, 264.96.2-24.g.2.30, 264.96.2-24.g.2.31, 264.96.2-24.g.2.32, 312.96.2-24.g.2.1, 312.96.2-24.g.2.2, 312.96.2-24.g.2.3, 312.96.2-24.g.2.4, 312.96.2-24.g.2.5, 312.96.2-24.g.2.6, 312.96.2-24.g.2.7, 312.96.2-24.g.2.8, 312.96.2-24.g.2.9, 312.96.2-24.g.2.10, 312.96.2-24.g.2.11, 312.96.2-24.g.2.12, 312.96.2-24.g.2.13, 312.96.2-24.g.2.14, 312.96.2-24.g.2.15, 312.96.2-24.g.2.16, 312.96.2-24.g.2.17, 312.96.2-24.g.2.18, 312.96.2-24.g.2.19, 312.96.2-24.g.2.20, 312.96.2-24.g.2.21, 312.96.2-24.g.2.22, 312.96.2-24.g.2.23, 312.96.2-24.g.2.24, 312.96.2-24.g.2.25, 312.96.2-24.g.2.26, 312.96.2-24.g.2.27, 312.96.2-24.g.2.28, 312.96.2-24.g.2.29, 312.96.2-24.g.2.30, 312.96.2-24.g.2.31, 312.96.2-24.g.2.32
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{12}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $2$
Newforms: 192.2.c.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} w + 3 x y w - x z w + y^{2} w - y z w - z^{2} w $
$=$ $2 x^{2} y + 3 x y^{2} - x y z + y^{3} - y^{2} z - y z^{2}$
$=$ $2 x^{3} + x^{2} y - x^{2} z - 2 x y^{2} - x z^{2} - y^{3} + y^{2} z + y z^{2}$
$=$ $2 x^{2} z + 3 x y z - x z^{2} + y^{2} z - y z^{2} - z^{3}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{3} y^{2} + 18 x^{2} y^{3} - 3 x^{2} y z^{2} - 18 x y^{2} z^{2} + x z^{4} + 4 y z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -4x^{5} - 10x^{4} + 10x^{2} + 4x $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:1)$, $(1:-2:1:0)$, $(-1:1:0:0)$, $(-1/2:1:0:0)$, $(-1/2:0:1:0)$, $(1:0:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{18}y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}w$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -2yz+\frac{1}{2}w^{2}$
$\displaystyle Y$ $=$ $\displaystyle 6y^{2}z^{3}w-\frac{9}{2}yz^{2}w^{3}+\frac{3}{4}zw^{5}$
$\displaystyle Z$ $=$ $\displaystyle yz$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^2}\cdot\frac{764330256xy^{9}-3567434400xy^{7}w^{2}+1486116072xy^{5}w^{4}-3468110580xy^{3}w^{6}+713557188xyw^{8}-89649504xz^{9}+880482312xz^{7}w^{2}-5829292008xz^{5}w^{4}+5971947492xz^{3}w^{6}-1606894494xzw^{8}+382124304y^{10}-1847507616y^{8}w^{2}+997732728y^{6}w^{4}-1666807956y^{4}w^{6}+604709712y^{2}w^{8}-107495424yz^{9}-15680533392yz^{7}w^{2}+4487527332yz^{5}w^{4}+9222537222yz^{3}w^{6}-298241874yzw^{8}-44719776z^{10}+460959336z^{8}w^{2}+5352054588z^{6}w^{4}-2239545378z^{4}w^{6}-1542713693z^{2}w^{8}+16384w^{10}}{w^{2}(42xy^{3}w^{4}+92xyw^{6}+137808xz^{7}+51300xz^{5}w^{2}+7788xz^{3}w^{4}+644xzw^{6}+42y^{4}w^{4}+92y^{2}w^{6}+103680yz^{7}+39672yz^{5}w^{2}+6474yz^{3}w^{4}+658yzw^{6}+69552z^{8}+30996z^{6}w^{2}+6126z^{4}w^{4}+523z^{2}w^{6})}$

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

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.96.3.bl.1 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.ct.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.ee.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.eh.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fp.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fr.2 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.fx.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.fz.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gm.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gm.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gn.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gn.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gq.2 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.gq.4 $24$ $2$ $2$ $3$ $1$ $1$
24.96.3.gr.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gr.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gu.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gu.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gv.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gv.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gy.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gy.4 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gz.2 $24$ $2$ $2$ $3$ $0$ $1$
24.96.3.gz.4 $24$ $2$ $2$ $3$ $0$ $1$
24.144.7.xq.1 $24$ $3$ $3$ $7$ $0$ $1\cdot2^{2}$
72.144.7.e.2 $72$ $3$ $3$ $7$ $?$ not computed
72.144.10.h.2 $72$ $3$ $3$ $10$ $?$ not computed
72.144.10.l.2 $72$ $3$ $3$ $10$ $?$ not computed
120.96.3.qy.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qz.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rc.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rd.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ro.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rp.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rs.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rt.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ti.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ti.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tj.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tj.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tm.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tm.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tn.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tn.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tq.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tq.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tr.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tr.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tu.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tu.4 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tv.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.tv.4 $120$ $2$ $2$ $3$ $?$ not computed
120.240.18.p.1 $120$ $5$ $5$ $18$ $?$ not computed
120.288.19.bfg.2 $120$ $6$ $6$ $19$ $?$ not computed
168.96.3.ok.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ol.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.oo.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.op.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pa.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pb.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pe.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.pf.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qu.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qu.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qv.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qv.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qy.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qy.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qz.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.qz.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rc.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rc.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rd.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rd.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rg.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rg.4 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rh.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.rh.4 $168$ $2$ $2$ $3$ $?$ not computed
264.96.3.ok.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ol.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.oo.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.op.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pa.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pb.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pe.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.pf.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qu.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qu.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qv.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qv.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qy.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qy.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qz.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.qz.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rc.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rc.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rd.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rd.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rg.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rg.4 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rh.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.rh.4 $264$ $2$ $2$ $3$ $?$ not computed
312.96.3.qy.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qz.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rc.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rd.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ro.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rp.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rs.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.rt.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ti.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ti.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tj.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tj.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tm.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tm.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tn.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tn.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tq.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tq.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tr.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tr.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tu.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tu.4 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tv.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.tv.4 $312$ $2$ $2$ $3$ $?$ not computed