↑

Properties

Label 24.144.5.gx.1
Level $24$
Index $144$
Genus $5$
Analytic rank $2$
Cusps $16$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $144$ $\PSL_2$-index:$144$
Genus: $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (of which $4$ are rational) Cusp widths $6^{8}\cdot12^{8}$ Cusp orbits $1^{4}\cdot2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12B5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.5.73

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\12&19\end{bmatrix}$, $\begin{bmatrix}5&9\\0&13\end{bmatrix}$, $\begin{bmatrix}13&3\\12&7\end{bmatrix}$, $\begin{bmatrix}13&15\\12&11\end{bmatrix}$, $\begin{bmatrix}19&0\\0&1\end{bmatrix}$, $\begin{bmatrix}23&0\\12&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.gx.1.1, 24.288.5-24.gx.1.2, 24.288.5-24.gx.1.3, 24.288.5-24.gx.1.4, 24.288.5-24.gx.1.5, 24.288.5-24.gx.1.6, 24.288.5-24.gx.1.7, 24.288.5-24.gx.1.8, 24.288.5-24.gx.1.9, 24.288.5-24.gx.1.10, 24.288.5-24.gx.1.11, 24.288.5-24.gx.1.12, 24.288.5-24.gx.1.13, 24.288.5-24.gx.1.14, 120.288.5-24.gx.1.1, 120.288.5-24.gx.1.2, 120.288.5-24.gx.1.3, 120.288.5-24.gx.1.4, 120.288.5-24.gx.1.5, 120.288.5-24.gx.1.6, 120.288.5-24.gx.1.7, 120.288.5-24.gx.1.8, 120.288.5-24.gx.1.9, 120.288.5-24.gx.1.10, 120.288.5-24.gx.1.11, 120.288.5-24.gx.1.12, 120.288.5-24.gx.1.13, 120.288.5-24.gx.1.14, 168.288.5-24.gx.1.1, 168.288.5-24.gx.1.2, 168.288.5-24.gx.1.3, 168.288.5-24.gx.1.4, 168.288.5-24.gx.1.5, 168.288.5-24.gx.1.6, 168.288.5-24.gx.1.7, 168.288.5-24.gx.1.8, 168.288.5-24.gx.1.9, 168.288.5-24.gx.1.10, 168.288.5-24.gx.1.11, 168.288.5-24.gx.1.12, 168.288.5-24.gx.1.13, 168.288.5-24.gx.1.14, 264.288.5-24.gx.1.1, 264.288.5-24.gx.1.2, 264.288.5-24.gx.1.3, 264.288.5-24.gx.1.4, 264.288.5-24.gx.1.5, 264.288.5-24.gx.1.6, 264.288.5-24.gx.1.7, 264.288.5-24.gx.1.8, 264.288.5-24.gx.1.9, 264.288.5-24.gx.1.10, 264.288.5-24.gx.1.11, 264.288.5-24.gx.1.12, 264.288.5-24.gx.1.13, 264.288.5-24.gx.1.14, 312.288.5-24.gx.1.1, 312.288.5-24.gx.1.2, 312.288.5-24.gx.1.3, 312.288.5-24.gx.1.4, 312.288.5-24.gx.1.5, 312.288.5-24.gx.1.6, 312.288.5-24.gx.1.7, 312.288.5-24.gx.1.8, 312.288.5-24.gx.1.9, 312.288.5-24.gx.1.10, 312.288.5-24.gx.1.11, 312.288.5-24.gx.1.12, 312.288.5-24.gx.1.13, 312.288.5-24.gx.1.14
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{26}\cdot3^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 36.2.a.a, 192.2.a.b, 576.2.a.b$^{2}$, 576.2.a.f

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ y w + y t + z^{2} $
$=$ $y^{2} + y t - z^{2} - w t$
$=$ $6 x^{2} + y^{2} + y w - y t + z^{2} - w^{2} - t^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ - 36 x^{4} y^{4} + 72 x^{3} y^{5} - 72 x^{3} y^{3} z^{2} - 144 x^{2} y^{6} + 240 x^{2} y^{4} z^{2} + \cdots + 5 z^{8} $
 Copy content Toggle raw display

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+w$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{4095yw^{17}+131076yw^{16}t+1859544yw^{15}t^{2}+15208704yw^{14}t^{3}+78104799yw^{13}t^{4}+255820140yw^{12}t^{5}+507716388yw^{11}t^{6}+476532144yw^{10}t^{7}-203475483yw^{9}t^{8}-1014930684yw^{8}t^{9}-863425800yw^{7}t^{10}+127612368yw^{6}t^{11}+658526418yw^{5}t^{12}+347950872yw^{4}t^{13}-45387432yw^{3}t^{14}-109602432yw^{2}t^{15}-40169529ywt^{16}-4969188yt^{17}-w^{18}-4095w^{17}t-118809w^{16}t^{2}-1515408w^{15}t^{3}-10969920w^{14}t^{4}-48548583w^{13}t^{5}-130446087w^{12}t^{6}-188005284w^{11}t^{7}-51788169w^{10}t^{8}+260851003w^{9}t^{9}+363404025w^{8}t^{10}+69611256w^{7}t^{11}-211122924w^{6}t^{12}-164871378w^{5}t^{13}-5556618w^{4}t^{14}+41601000w^{3}t^{15}+18221283w^{2}t^{16}+2484585wt^{17}-t^{18}}{t^{3}(w+t)^{6}(512yw^{8}+8704yw^{7}t+57344yw^{6}t^{2}+191809yw^{5}t^{3}+359402yw^{4}t^{4}+390835yw^{3}t^{5}+243866yw^{2}t^{6}+80752ywt^{7}+10976yt^{8}-512w^{8}t-7168w^{7}t^{2}-37375w^{6}t^{3}-95059w^{5}t^{4}-129362w^{4}t^{5}-95767w^{3}t^{6}-36260w^{2}t^{7}-5488wt^{8})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

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
12.72.1.f.1 $12$ $2$ $2$ $1$ $0$ $1^{4}$
24.48.1.ik.1 $24$ $3$ $3$ $1$ $1$ $1^{4}$
24.72.1.bn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.cd.1 $24$ $2$ $2$ $1$ $1$ $1^{4}$
24.72.3.mq.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.qj.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.qp.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.72.3.tj.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.9.bi.1 $24$ $2$ $2$ $9$ $2$ $2^{2}$
24.288.9.bi.2 $24$ $2$ $2$ $9$ $2$ $2^{2}$
24.288.13.mk.1 $24$ $2$ $2$ $13$ $3$ $1^{8}$
24.288.13.mm.1 $24$ $2$ $2$ $13$ $3$ $1^{8}$
24.288.13.oc.1 $24$ $2$ $2$ $13$ $2$ $2^{4}$
24.288.13.od.1 $24$ $2$ $2$ $13$ $2$ $2^{4}$
24.288.13.og.1 $24$ $2$ $2$ $13$ $4$ $1^{8}$
24.288.13.oi.1 $24$ $2$ $2$ $13$ $3$ $1^{8}$
72.432.21.mh.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.pf.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.tc.1 $72$ $3$ $3$ $21$ $?$ not computed
120.288.9.fw.1 $120$ $2$ $2$ $9$ $?$ not computed
120.288.9.fw.2 $120$ $2$ $2$ $9$ $?$ not computed
120.288.13.eea.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.eeb.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.efg.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.efh.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.efk.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.efl.1 $120$ $2$ $2$ $13$ $?$ not computed
168.288.9.dm.1 $168$ $2$ $2$ $9$ $?$ not computed
168.288.9.dm.2 $168$ $2$ $2$ $9$ $?$ not computed
168.288.13.cfe.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cff.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cgk.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cgl.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cgo.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cgp.1 $168$ $2$ $2$ $13$ $?$ not computed
264.288.9.dm.1 $264$ $2$ $2$ $9$ $?$ not computed
264.288.9.dm.2 $264$ $2$ $2$ $9$ $?$ not computed
264.288.13.cck.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.ccl.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.cdq.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.cdr.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.cdu.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.cdv.1 $264$ $2$ $2$ $13$ $?$ not computed
312.288.9.dm.1 $312$ $2$ $2$ $9$ $?$ not computed
312.288.9.dm.2 $312$ $2$ $2$ $9$ $?$ not computed
312.288.13.cfu.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.cfv.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.cha.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.chb.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.che.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.chf.1 $312$ $2$ $2$ $13$ $?$ not computed