↑

Properties

Label 24.144.5.fv.1
Level $24$
Index $144$
Genus $5$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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: $16$ (none of which are rational) Cusp widths $6^{8}\cdot12^{8}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12B5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.5.78

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\6&7\end{bmatrix}$, $\begin{bmatrix}5&3\\6&5\end{bmatrix}$, $\begin{bmatrix}7&9\\18&11\end{bmatrix}$, $\begin{bmatrix}19&6\\12&23\end{bmatrix}$, $\begin{bmatrix}23&15\\18&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.5-24.fv.1.1, 24.288.5-24.fv.1.2, 24.288.5-24.fv.1.3, 24.288.5-24.fv.1.4, 24.288.5-24.fv.1.5, 24.288.5-24.fv.1.6, 24.288.5-24.fv.1.7, 24.288.5-24.fv.1.8, 120.288.5-24.fv.1.1, 120.288.5-24.fv.1.2, 120.288.5-24.fv.1.3, 120.288.5-24.fv.1.4, 120.288.5-24.fv.1.5, 120.288.5-24.fv.1.6, 120.288.5-24.fv.1.7, 120.288.5-24.fv.1.8, 168.288.5-24.fv.1.1, 168.288.5-24.fv.1.2, 168.288.5-24.fv.1.3, 168.288.5-24.fv.1.4, 168.288.5-24.fv.1.5, 168.288.5-24.fv.1.6, 168.288.5-24.fv.1.7, 168.288.5-24.fv.1.8, 264.288.5-24.fv.1.1, 264.288.5-24.fv.1.2, 264.288.5-24.fv.1.3, 264.288.5-24.fv.1.4, 264.288.5-24.fv.1.5, 264.288.5-24.fv.1.6, 264.288.5-24.fv.1.7, 264.288.5-24.fv.1.8, 312.288.5-24.fv.1.1, 312.288.5-24.fv.1.2, 312.288.5-24.fv.1.3, 312.288.5-24.fv.1.4, 312.288.5-24.fv.1.5, 312.288.5-24.fv.1.6, 312.288.5-24.fv.1.7, 312.288.5-24.fv.1.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{28}\cdot3^{9}$
Simple: no
Squarefree: no
Decomposition: $1^{5}$
Newforms: 144.2.a.a, 192.2.a.d, 576.2.a.d$^{2}$, 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 no $\Q_p$ points for $p=7,31$, and therefore no rational points.

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.d.1 $12$ $2$ $2$ $1$ $0$ $1^{4}$
24.48.1.ie.1 $24$ $3$ $3$ $1$ $0$ $1^{4}$
24.72.1.bn.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.1.cg.1 $24$ $2$ $2$ $1$ $0$ $1^{4}$
24.72.3.lo.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.qd.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.qm.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.tk.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.13.nm.1 $24$ $2$ $2$ $13$ $0$ $2^{4}$
24.288.13.nn.1 $24$ $2$ $2$ $13$ $0$ $2^{4}$
72.432.21.lv.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.od.1 $72$ $3$ $3$ $21$ $?$ not computed
72.432.21.ru.1 $72$ $3$ $3$ $21$ $?$ not computed
120.288.13.ecm.1 $120$ $2$ $2$ $13$ $?$ not computed
120.288.13.ecn.1 $120$ $2$ $2$ $13$ $?$ not computed
168.288.13.cdq.1 $168$ $2$ $2$ $13$ $?$ not computed
168.288.13.cdr.1 $168$ $2$ $2$ $13$ $?$ not computed
264.288.13.caw.1 $264$ $2$ $2$ $13$ $?$ not computed
264.288.13.cax.1 $264$ $2$ $2$ $13$ $?$ not computed
312.288.13.ceg.1 $312$ $2$ $2$ $13$ $?$ not computed
312.288.13.ceh.1 $312$ $2$ $2$ $13$ $?$ not computed