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Properties

Label 24.48.2.g.1
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: 24.48.2.13

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&9\\16&17\end{bmatrix}$, $\begin{bmatrix}11&15\\12&13\end{bmatrix}$, $\begin{bmatrix}13&21\\20&1\end{bmatrix}$, $\begin{bmatrix}17&6\\8&11\end{bmatrix}$, $\begin{bmatrix}23&15\\20&7\end{bmatrix}$, $\begin{bmatrix}23&21\\12&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.2-24.g.1.1, 24.96.2-24.g.1.2, 24.96.2-24.g.1.3, 24.96.2-24.g.1.4, 24.96.2-24.g.1.5, 24.96.2-24.g.1.6, 24.96.2-24.g.1.7, 24.96.2-24.g.1.8, 24.96.2-24.g.1.9, 24.96.2-24.g.1.10, 24.96.2-24.g.1.11, 24.96.2-24.g.1.12, 24.96.2-24.g.1.13, 24.96.2-24.g.1.14, 24.96.2-24.g.1.15, 24.96.2-24.g.1.16, 24.96.2-24.g.1.17, 24.96.2-24.g.1.18, 24.96.2-24.g.1.19, 24.96.2-24.g.1.20, 24.96.2-24.g.1.21, 24.96.2-24.g.1.22, 24.96.2-24.g.1.23, 24.96.2-24.g.1.24, 24.96.2-24.g.1.25, 24.96.2-24.g.1.26, 24.96.2-24.g.1.27, 24.96.2-24.g.1.28, 24.96.2-24.g.1.29, 24.96.2-24.g.1.30, 24.96.2-24.g.1.31, 24.96.2-24.g.1.32, 120.96.2-24.g.1.1, 120.96.2-24.g.1.2, 120.96.2-24.g.1.3, 120.96.2-24.g.1.4, 120.96.2-24.g.1.5, 120.96.2-24.g.1.6, 120.96.2-24.g.1.7, 120.96.2-24.g.1.8, 120.96.2-24.g.1.9, 120.96.2-24.g.1.10, 120.96.2-24.g.1.11, 120.96.2-24.g.1.12, 120.96.2-24.g.1.13, 120.96.2-24.g.1.14, 120.96.2-24.g.1.15, 120.96.2-24.g.1.16, 120.96.2-24.g.1.17, 120.96.2-24.g.1.18, 120.96.2-24.g.1.19, 120.96.2-24.g.1.20, 120.96.2-24.g.1.21, 120.96.2-24.g.1.22, 120.96.2-24.g.1.23, 120.96.2-24.g.1.24, 120.96.2-24.g.1.25, 120.96.2-24.g.1.26, 120.96.2-24.g.1.27, 120.96.2-24.g.1.28, 120.96.2-24.g.1.29, 120.96.2-24.g.1.30, 120.96.2-24.g.1.31, 120.96.2-24.g.1.32, 168.96.2-24.g.1.1, 168.96.2-24.g.1.2, 168.96.2-24.g.1.3, 168.96.2-24.g.1.4, 168.96.2-24.g.1.5, 168.96.2-24.g.1.6, 168.96.2-24.g.1.7, 168.96.2-24.g.1.8, 168.96.2-24.g.1.9, 168.96.2-24.g.1.10, 168.96.2-24.g.1.11, 168.96.2-24.g.1.12, 168.96.2-24.g.1.13, 168.96.2-24.g.1.14, 168.96.2-24.g.1.15, 168.96.2-24.g.1.16, 168.96.2-24.g.1.17, 168.96.2-24.g.1.18, 168.96.2-24.g.1.19, 168.96.2-24.g.1.20, 168.96.2-24.g.1.21, 168.96.2-24.g.1.22, 168.96.2-24.g.1.23, 168.96.2-24.g.1.24, 168.96.2-24.g.1.25, 168.96.2-24.g.1.26, 168.96.2-24.g.1.27, 168.96.2-24.g.1.28, 168.96.2-24.g.1.29, 168.96.2-24.g.1.30, 168.96.2-24.g.1.31, 168.96.2-24.g.1.32, 264.96.2-24.g.1.1, 264.96.2-24.g.1.2, 264.96.2-24.g.1.3, 264.96.2-24.g.1.4, 264.96.2-24.g.1.5, 264.96.2-24.g.1.6, 264.96.2-24.g.1.7, 264.96.2-24.g.1.8, 264.96.2-24.g.1.9, 264.96.2-24.g.1.10, 264.96.2-24.g.1.11, 264.96.2-24.g.1.12, 264.96.2-24.g.1.13, 264.96.2-24.g.1.14, 264.96.2-24.g.1.15, 264.96.2-24.g.1.16, 264.96.2-24.g.1.17, 264.96.2-24.g.1.18, 264.96.2-24.g.1.19, 264.96.2-24.g.1.20, 264.96.2-24.g.1.21, 264.96.2-24.g.1.22, 264.96.2-24.g.1.23, 264.96.2-24.g.1.24, 264.96.2-24.g.1.25, 264.96.2-24.g.1.26, 264.96.2-24.g.1.27, 264.96.2-24.g.1.28, 264.96.2-24.g.1.29, 264.96.2-24.g.1.30, 264.96.2-24.g.1.31, 264.96.2-24.g.1.32, 312.96.2-24.g.1.1, 312.96.2-24.g.1.2, 312.96.2-24.g.1.3, 312.96.2-24.g.1.4, 312.96.2-24.g.1.5, 312.96.2-24.g.1.6, 312.96.2-24.g.1.7, 312.96.2-24.g.1.8, 312.96.2-24.g.1.9, 312.96.2-24.g.1.10, 312.96.2-24.g.1.11, 312.96.2-24.g.1.12, 312.96.2-24.g.1.13, 312.96.2-24.g.1.14, 312.96.2-24.g.1.15, 312.96.2-24.g.1.16, 312.96.2-24.g.1.17, 312.96.2-24.g.1.18, 312.96.2-24.g.1.19, 312.96.2-24.g.1.20, 312.96.2-24.g.1.21, 312.96.2-24.g.1.22, 312.96.2-24.g.1.23, 312.96.2-24.g.1.24, 312.96.2-24.g.1.25, 312.96.2-24.g.1.26, 312.96.2-24.g.1.27, 312.96.2-24.g.1.28, 312.96.2-24.g.1.29, 312.96.2-24.g.1.30, 312.96.2-24.g.1.31, 312.96.2-24.g.1.32
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 $ $=$ $ x^{2} y + x^{2} z + 2 x^{2} w + 2 y z w + 2 z^{2} w + 2 z w^{2} $
$=$ $2 y^{2} z + 3 y z^{2} - y z w + z^{3} - z^{2} w - z w^{2}$
$=$ $2 y^{3} + y^{2} z - y^{2} w - 2 y z^{2} - y w^{2} - z^{3} + z^{2} w + z w^{2}$
$=$ $2 y^{2} w + 3 y z w - y w^{2} + z^{2} w - z w^{2} - w^{3}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} y + 18 x^{4} z + 3 x^{2} y^{2} z + 18 x^{2} y z^{2} + y^{3} z^{2} + 4 y^{2} z^{3} $
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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
$(1:0:0:0)$, $(0:1:-2:1)$, $(0:-1:1:0)$, $(0:-1/2:1:0)$, $(0:-1/2:0:1)$, $(0:1:0:1)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 2z$
$\displaystyle Z$ $=$ $\displaystyle w$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle \frac{1}{2}x^{2}$
$\displaystyle Y$ $=$ $\displaystyle -\frac{9}{4}x^{5}w-\frac{9}{2}x^{3}zw^{2}-2xz^{2}w^{3}$
$\displaystyle Z$ $=$ $\displaystyle x^{2}+zw$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^2}\cdot\frac{1024x^{10}-512x^{8}w^{2}+27648x^{6}w^{4}+503424x^{4}w^{6}+13535424x^{2}w^{8}+589842yz^{9}+2064609yz^{8}w+2213244yz^{7}w^{2}+15106338yz^{6}w^{3}-18725580yz^{5}w^{4}+44517908yz^{4}w^{5}-36568084yz^{3}w^{6}+5716766yz^{2}w^{7}+5471154yzw^{8}+13851yw^{9}+294930z^{10}+737505z^{9}w-514674z^{8}w^{2}+4968945z^{7}w^{3}-16512579z^{6}w^{4}+17654067z^{5}w^{5}-12221775z^{4}w^{6}-15215829z^{3}w^{7}+9102381z^{2}w^{8}+11767392zw^{9}+6885w^{10}}{w^{3}(576x^{2}w^{5}-2yz^{6}-13yz^{5}w-54yz^{4}w^{2}-102yz^{3}w^{3}-204yz^{2}w^{4}+171yzw^{5}-2z^{7}-13z^{6}w-56z^{5}w^{2}-113z^{4}w^{3}-243z^{3}w^{4}+126z^{2}w^{5}+469zw^{6})}$

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