↑

Properties

Label 24.192.5.br.2
Level $24$
Index $192$
Genus $5$
Analytic rank $1$
Cusps $24$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $192$ $\PSL_2$-index:$192$
Genus: $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (none of which are rational) Cusp widths $4^{12}\cdot12^{12}$ Cusp orbits $2^{4}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12E5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.192.5.281

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&22\\6&17\end{bmatrix}$, $\begin{bmatrix}19&4\\0&1\end{bmatrix}$, $\begin{bmatrix}19&12\\18&5\end{bmatrix}$, $\begin{bmatrix}23&2\\12&11\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^5.D_6$
Contains $-I$: yes
Quadratic refinements: 24.384.5-24.br.2.1, 24.384.5-24.br.2.2, 24.384.5-24.br.2.3, 24.384.5-24.br.2.4, 24.384.5-24.br.2.5, 24.384.5-24.br.2.6, 24.384.5-24.br.2.7, 24.384.5-24.br.2.8, 120.384.5-24.br.2.1, 120.384.5-24.br.2.2, 120.384.5-24.br.2.3, 120.384.5-24.br.2.4, 120.384.5-24.br.2.5, 120.384.5-24.br.2.6, 120.384.5-24.br.2.7, 120.384.5-24.br.2.8, 168.384.5-24.br.2.1, 168.384.5-24.br.2.2, 168.384.5-24.br.2.3, 168.384.5-24.br.2.4, 168.384.5-24.br.2.5, 168.384.5-24.br.2.6, 168.384.5-24.br.2.7, 168.384.5-24.br.2.8, 264.384.5-24.br.2.1, 264.384.5-24.br.2.2, 264.384.5-24.br.2.3, 264.384.5-24.br.2.4, 264.384.5-24.br.2.5, 264.384.5-24.br.2.6, 264.384.5-24.br.2.7, 264.384.5-24.br.2.8, 312.384.5-24.br.2.1, 312.384.5-24.br.2.2, 312.384.5-24.br.2.3, 312.384.5-24.br.2.4, 312.384.5-24.br.2.5, 312.384.5-24.br.2.6, 312.384.5-24.br.2.7, 312.384.5-24.br.2.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $384$

Jacobian

Conductor: $2^{27}\cdot3^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}\cdot2$
Newforms: 72.2.a.a, 192.2.a.d, 192.2.c.a, 576.2.a.b

Models

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

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

Singular plane model Singular plane model

$ 0 $ $=$ $ 16 x^{8} - 24 x^{7} y + 36 x^{6} y^{2} - 36 x^{6} z^{2} - 16 x^{5} y^{3} + 6 x^{5} y z^{2} + \cdots + 81 z^{8} $
 Copy content Toggle raw display

Rational points

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

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}t$

Maps to other modular curves

$j$-invariant map of degree 192 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^2\cdot3^3}\cdot\frac{543881998203453yw^{23}-127796464032660yw^{21}t^{2}+1270536541551855yw^{19}t^{4}-292215558607056yw^{17}t^{6}+1026959015156436yw^{15}t^{8}-229765933310544yw^{13}t^{10}+321451521469209yw^{11}t^{12}-68572938360168yw^{9}t^{14}+26478498435291yw^{7}t^{16}-4039299493596yw^{5}t^{18}+395268490863yw^{3}t^{20}+719232188919426z^{2}w^{22}-29720956613490z^{2}w^{20}t^{2}+1641558140971740z^{2}w^{18}t^{4}-81917351498826z^{2}w^{16}t^{6}+1303431035750424z^{2}w^{14}t^{8}-78951812553144z^{2}w^{12}t^{10}+390396041624304z^{2}w^{10}t^{12}-27512556156558z^{2}w^{8}t^{14}+26970360643350z^{2}w^{6}t^{16}-2698457738382z^{2}w^{4}t^{18}+600558711012z^{2}w^{2}t^{20}-45344587578z^{2}t^{22}-1263114187122879zw^{23}-23776578755001zw^{21}t^{2}-2898720828395289zw^{19}t^{4}-44921417883327zw^{17}t^{6}-2299792106379204zw^{15}t^{8}-19814688586620zw^{13}t^{10}-689979427725339zw^{11}t^{12}+994044587967zw^{9}t^{14}-47280131295921zw^{7}t^{16}+1307657439009zw^{5}t^{18}-704921637969zw^{3}t^{20}+17048369553zwt^{22}+723690239143554w^{24}-225875252568564w^{22}t^{2}+1706943499378254w^{20}t^{4}-524342535477120w^{18}t^{6}+1412364977686152w^{16}t^{8}-422147563002108w^{14}t^{10}+467513235230418w^{12}t^{12}-132653538165888w^{10}t^{14}+50343409944846w^{8}t^{16}-11943704143476w^{6}t^{18}+2569744811214w^{4}t^{20}-412316860416w^{2}t^{22}+34359738368t^{24}}{t^{4}(15479282007yw^{19}-3439486152yw^{17}t^{2}+46900215yw^{15}t^{4}-20286612yw^{13}t^{6}+15394941yw^{11}t^{8}-1264248yw^{9}t^{10}-205470yw^{7}t^{12}+36408yw^{5}t^{14}-1665yw^{3}t^{16}+20639160774z^{2}w^{18}-1146849678z^{2}w^{16}t^{2}-509098608z^{2}w^{14}t^{4}-103107330z^{2}w^{12}t^{6}+22935366z^{2}w^{10}t^{8}+5218830z^{2}w^{8}t^{10}-1674072z^{2}w^{6}t^{12}+192916z^{2}w^{4}t^{14}-12834z^{2}w^{2}t^{16}+450z^{2}t^{18}-36118442781zw^{19}-573424839zw^{17}t^{2}+748743507zw^{15}t^{4}+298790613zw^{13}t^{6}-18151587zw^{11}t^{8}-12441573zw^{9}t^{10}+1014642zw^{7}t^{12}+213206zw^{5}t^{14}-36453zw^{3}t^{16}+1665zwt^{18}+20639160774w^{20}-6306669396w^{18}t^{2}+637751070w^{16}t^{4}-71764704w^{14}t^{6}+18097938w^{12}t^{8}-2075544w^{10}t^{10}+9252w^{8}t^{12}+14976w^{6}t^{14}-810w^{4}t^{16})}$

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
12.96.1.d.1 $12$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.96.1.cj.2 $24$ $2$ $2$ $1$ $0$ $1^{2}\cdot2$
24.96.1.cp.2 $24$ $2$ $2$ $1$ $1$ $1^{2}\cdot2$
24.96.3.be.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.bt.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cb.2 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cc.2 $24$ $2$ $2$ $3$ $1$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.576.25.bk.1 $24$ $3$ $3$ $25$ $3$ $1^{10}\cdot2^{5}$