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Properties

Label 24.192.3-24.er.1.3
Level $24$
Index $192$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $192$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $2^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24V3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.3.180

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&12\\0&5\end{bmatrix}$, $\begin{bmatrix}5&12\\0&13\end{bmatrix}$, $\begin{bmatrix}11&16\\12&7\end{bmatrix}$, $\begin{bmatrix}17&1\\0&7\end{bmatrix}$, $\begin{bmatrix}23&2\\12&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_4:D_4\times D_6$
Contains $-I$: no $\quad$ (see 24.96.3.er.1 for the level structure with $-I$)
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $8$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{16}\cdot3^{3}$
Simple: no
Squarefree: no
Decomposition: $1^{3}$
Newforms: 48.2.a.a, 192.2.a.d$^{2}$

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

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

$ 0 $ $=$ $ 9025 x^{8} - 181672 x^{6} y^{2} - 14820 x^{6} y z - 19000 x^{6} z^{2} + 348376 x^{4} y^{4} + \cdots + 121 z^{8} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ 3x^{8} + 28x^{6} - 14x^{4} + 28x^{2} + 3 $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^5\,\frac{1593968xt^{10}u+3305504xt^{8}u^{3}+1622284xt^{6}u^{5}+197322xt^{4}u^{7}+114xt^{2}u^{9}-3077xu^{11}-209664yzt^{10}-963072yzt^{8}u^{2}-487872yzt^{6}u^{4}-160416yzt^{4}u^{6}+26208yzt^{2}u^{8}-13104yzu^{10}+153232yt^{10}u-379424yt^{8}u^{3}-711244yt^{6}u^{5}+288438yt^{4}u^{7}-43794yt^{2}u^{9}+3077yu^{11}-61152z^{2}t^{10}-280896z^{2}t^{8}u^{2}-142296z^{2}t^{6}u^{4}-46788z^{2}t^{4}u^{6}+7644z^{2}t^{2}u^{8}-3822z^{2}u^{10}-291200zwt^{9}u-487680zwt^{7}u^{3}-151840zwt^{5}u^{5}-80960zwt^{3}u^{7}+7280zwtu^{9}+728000zt^{10}u+1219200zt^{8}u^{3}+379600zt^{6}u^{5}+202400zt^{4}u^{7}-18200zt^{2}u^{9}+157248wt^{11}+1159104wt^{9}u^{2}+1097424wt^{7}u^{4}+348072wt^{5}u^{6}+101784wt^{3}u^{8}-1092wtu^{10}+27120t^{12}+93568t^{10}u^{2}-150516t^{8}u^{4}-171766t^{6}u^{6}-124658t^{4}u^{8}-1621t^{2}u^{10}+1038u^{12}}{u^{2}(334xt^{8}u+1029xt^{6}u^{3}+1481xt^{4}u^{5}+540xt^{2}u^{7}+45xu^{9}+288yzt^{8}+1008yzt^{6}u^{2}+432yzt^{4}u^{4}-2254yt^{8}u-4149yt^{6}u^{3}-2201yt^{4}u^{5}-540yt^{2}u^{7}-45yu^{9}+84z^{2}t^{8}+294z^{2}t^{6}u^{2}+126z^{2}t^{4}u^{4}+320zwt^{7}u+520zwt^{5}u^{3}+120zwt^{3}u^{5}-800zt^{8}u-1300zt^{6}u^{3}-300zt^{4}u^{5}-216wt^{9}-1236wt^{7}u^{2}-1104wt^{5}u^{4}-180wt^{3}u^{6}+390t^{10}+2129t^{8}u^{2}+2059t^{6}u^{4}+636t^{4}u^{6}+45t^{2}u^{8})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.er.1 :

$\displaystyle X$ $=$ $\displaystyle t$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle u$

Equation of the image curve:

$0$ $=$ $ 9025X^{8}-181672X^{6}Y^{2}+348376X^{4}Y^{4}-42912X^{2}Y^{6}+1296Y^{8}-14820X^{6}YZ-170760X^{4}Y^{3}Z+134352X^{2}Y^{5}Z-7776Y^{7}Z-19000X^{6}Z^{2}+24270X^{4}Y^{2}Z^{2}-81016X^{2}Y^{4}Z^{2}+15480Y^{6}Z^{2}-49800X^{4}YZ^{3}+76620X^{2}Y^{3}Z^{3}-15768Y^{5}Z^{3}+4490X^{4}Z^{4}-41520X^{2}Y^{2}Z^{4}+16833Y^{4}Z^{4}+4428X^{2}YZ^{5}-8784Y^{3}Z^{5}+40X^{2}Z^{6}+2770Y^{2}Z^{6}-792YZ^{7}+121Z^{8} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.er.1 :

$\displaystyle X$ $=$ $\displaystyle -2w^{3}+4w^{2}t+w^{2}u+wu^{2}-2tu^{2}$
$\displaystyle Y$ $=$ $\displaystyle -72zw^{8}u^{3}+144zw^{7}tu^{3}+96zw^{6}tu^{4}+60zw^{6}u^{5}-72zw^{5}tu^{5}-48zw^{4}tu^{6}-12zw^{4}u^{7}-88w^{9}u^{3}+176w^{8}tu^{3}+22w^{8}u^{4}+88w^{7}tu^{4}+104w^{7}u^{5}-120w^{6}tu^{5}+18w^{6}u^{6}-60w^{5}tu^{6}-16w^{5}u^{7}+16w^{4}tu^{7}-4w^{4}u^{8}+8w^{3}tu^{8}$
$\displaystyle Z$ $=$ $\displaystyle w^{2}u+wu^{2}$

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $48$ $24$ $0$ $0$ full Jacobian
8.48.0-8.x.1.2 $8$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0-8.x.1.2 $8$ $4$ $4$ $0$ $0$ full Jacobian
24.96.1-12.h.1.1 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.96.1-12.h.1.26 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.96.1-24.ix.1.6 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.96.1-24.ix.1.11 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.96.1-24.ix.1.22 $24$ $2$ $2$ $1$ $0$ $1^{2}$
24.96.1-24.ix.1.27 $24$ $2$ $2$ $1$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.384.5-24.ep.1.1 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.ep.1.7 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.ep.2.1 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.ep.2.7 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.eq.1.1 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.eq.1.7 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.eq.2.1 $24$ $2$ $2$ $5$ $0$ $2$
24.384.5-24.eq.2.7 $24$ $2$ $2$ $5$ $0$ $2$
24.384.9-24.ls.1.13 $24$ $2$ $2$ $9$ $1$ $1^{6}$
24.384.9-24.lt.1.1 $24$ $2$ $2$ $9$ $0$ $1^{6}$
24.384.9-24.lu.1.7 $24$ $2$ $2$ $9$ $2$ $1^{6}$
24.384.9-24.lv.1.1 $24$ $2$ $2$ $9$ $2$ $1^{6}$
24.384.9-24.lw.1.1 $24$ $2$ $2$ $9$ $0$ $2\cdot4$
24.384.9-24.lw.2.1 $24$ $2$ $2$ $9$ $0$ $2\cdot4$
24.384.9-24.lx.1.1 $24$ $2$ $2$ $9$ $0$ $2\cdot4$
24.384.9-24.lx.2.1 $24$ $2$ $2$ $9$ $0$ $2\cdot4$
24.576.13-24.jk.1.1 $24$ $3$ $3$ $13$ $2$ $1^{10}$
48.384.7-48.dp.1.3 $48$ $2$ $2$ $7$ $2$ $1^{4}$
48.384.7-48.dq.1.3 $48$ $2$ $2$ $7$ $4$ $1^{4}$
48.384.7-48.dr.1.1 $48$ $2$ $2$ $7$ $0$ $4$
48.384.7-48.dr.1.13 $48$ $2$ $2$ $7$ $0$ $4$
48.384.7-48.dr.2.1 $48$ $2$ $2$ $7$ $0$ $4$
48.384.7-48.dr.2.13 $48$ $2$ $2$ $7$ $0$ $4$
48.384.7-48.ds.1.3 $48$ $2$ $2$ $7$ $0$ $1^{4}$
48.384.7-48.dt.1.3 $48$ $2$ $2$ $7$ $2$ $1^{4}$
48.384.11-48.t.1.3 $48$ $2$ $2$ $11$ $0$ $1^{4}\cdot2^{2}$
48.384.11-48.ba.1.3 $48$ $2$ $2$ $11$ $8$ $1^{4}\cdot2^{2}$
48.384.11-48.bi.1.3 $48$ $2$ $2$ $11$ $0$ $4^{2}$
48.384.11-48.bi.1.13 $48$ $2$ $2$ $11$ $0$ $4^{2}$
48.384.11-48.bi.2.3 $48$ $2$ $2$ $11$ $0$ $4^{2}$
48.384.11-48.bi.2.13 $48$ $2$ $2$ $11$ $0$ $4^{2}$
48.384.11-48.bk.1.3 $48$ $2$ $2$ $11$ $0$ $1^{4}\cdot2^{2}$
48.384.11-48.bn.1.3 $48$ $2$ $2$ $11$ $4$ $1^{4}\cdot2^{2}$
120.384.5-120.vf.1.3 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vf.1.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vf.2.3 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vf.2.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vg.1.4 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vg.1.13 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vg.2.4 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.vg.2.13 $120$ $2$ $2$ $5$ $?$ not computed
120.384.9-120.blo.1.9 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.blp.1.5 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.blq.1.9 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.blr.1.5 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.bls.1.3 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.bls.2.3 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.blt.1.5 $120$ $2$ $2$ $9$ $?$ not computed
120.384.9-120.blt.2.5 $120$ $2$ $2$ $9$ $?$ not computed
168.384.5-168.vf.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vf.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vf.2.2 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vf.2.13 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vg.1.1 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vg.1.15 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vg.2.3 $168$ $2$ $2$ $5$ $?$ not computed
168.384.5-168.vg.2.13 $168$ $2$ $2$ $5$ $?$ not computed
168.384.9-168.bjx.1.3 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bjy.1.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bjz.1.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bka.1.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bkb.1.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bkb.2.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bkc.1.1 $168$ $2$ $2$ $9$ $?$ not computed
168.384.9-168.bkc.2.1 $168$ $2$ $2$ $9$ $?$ not computed
240.384.7-240.pr.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ps.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pt.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pt.1.29 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pt.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pt.2.29 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pu.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.pv.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.11-240.bp.1.1 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.bs.1.1 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cc.1.3 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cc.1.57 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cc.2.3 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cc.2.57 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cg.1.1 $240$ $2$ $2$ $11$ $?$ not computed
240.384.11-240.cj.1.1 $240$ $2$ $2$ $11$ $?$ not computed
264.384.5-264.vf.1.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vf.1.15 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vf.2.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vf.2.15 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vg.1.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vg.1.15 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vg.2.1 $264$ $2$ $2$ $5$ $?$ not computed
264.384.5-264.vg.2.15 $264$ $2$ $2$ $5$ $?$ not computed
264.384.9-264.bjc.1.15 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjd.1.1 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bje.1.15 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjf.1.1 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjg.1.1 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjg.2.1 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjh.1.1 $264$ $2$ $2$ $9$ $?$ not computed
264.384.9-264.bjh.2.1 $264$ $2$ $2$ $9$ $?$ not computed
312.384.5-312.vf.1.2 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vf.1.11 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vf.2.7 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vf.2.9 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vg.1.5 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vg.1.10 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vg.2.6 $312$ $2$ $2$ $5$ $?$ not computed
312.384.5-312.vg.2.9 $312$ $2$ $2$ $5$ $?$ not computed
312.384.9-312.blo.1.5 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.blp.1.6 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.blq.1.9 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.blr.1.11 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.bls.1.5 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.bls.2.9 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.blt.1.7 $312$ $2$ $2$ $9$ $?$ not computed
312.384.9-312.blt.2.3 $312$ $2$ $2$ $9$ $?$ not computed