↑

Properties

Label 24.48.1.x.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.334

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&0\\12&5\end{bmatrix}$, $\begin{bmatrix}17&18\\4&17\end{bmatrix}$, $\begin{bmatrix}19&2\\0&23\end{bmatrix}$, $\begin{bmatrix}19&2\\12&11\end{bmatrix}$, $\begin{bmatrix}21&22\\4&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.x.1.1, 24.96.1-24.x.1.2, 24.96.1-24.x.1.3, 24.96.1-24.x.1.4, 24.96.1-24.x.1.5, 24.96.1-24.x.1.6, 24.96.1-24.x.1.7, 24.96.1-24.x.1.8, 24.96.1-24.x.1.9, 24.96.1-24.x.1.10, 24.96.1-24.x.1.11, 24.96.1-24.x.1.12, 24.96.1-24.x.1.13, 24.96.1-24.x.1.14, 24.96.1-24.x.1.15, 24.96.1-24.x.1.16, 120.96.1-24.x.1.1, 120.96.1-24.x.1.2, 120.96.1-24.x.1.3, 120.96.1-24.x.1.4, 120.96.1-24.x.1.5, 120.96.1-24.x.1.6, 120.96.1-24.x.1.7, 120.96.1-24.x.1.8, 120.96.1-24.x.1.9, 120.96.1-24.x.1.10, 120.96.1-24.x.1.11, 120.96.1-24.x.1.12, 120.96.1-24.x.1.13, 120.96.1-24.x.1.14, 120.96.1-24.x.1.15, 120.96.1-24.x.1.16, 168.96.1-24.x.1.1, 168.96.1-24.x.1.2, 168.96.1-24.x.1.3, 168.96.1-24.x.1.4, 168.96.1-24.x.1.5, 168.96.1-24.x.1.6, 168.96.1-24.x.1.7, 168.96.1-24.x.1.8, 168.96.1-24.x.1.9, 168.96.1-24.x.1.10, 168.96.1-24.x.1.11, 168.96.1-24.x.1.12, 168.96.1-24.x.1.13, 168.96.1-24.x.1.14, 168.96.1-24.x.1.15, 168.96.1-24.x.1.16, 264.96.1-24.x.1.1, 264.96.1-24.x.1.2, 264.96.1-24.x.1.3, 264.96.1-24.x.1.4, 264.96.1-24.x.1.5, 264.96.1-24.x.1.6, 264.96.1-24.x.1.7, 264.96.1-24.x.1.8, 264.96.1-24.x.1.9, 264.96.1-24.x.1.10, 264.96.1-24.x.1.11, 264.96.1-24.x.1.12, 264.96.1-24.x.1.13, 264.96.1-24.x.1.14, 264.96.1-24.x.1.15, 264.96.1-24.x.1.16, 312.96.1-24.x.1.1, 312.96.1-24.x.1.2, 312.96.1-24.x.1.3, 312.96.1-24.x.1.4, 312.96.1-24.x.1.5, 312.96.1-24.x.1.6, 312.96.1-24.x.1.7, 312.96.1-24.x.1.8, 312.96.1-24.x.1.9, 312.96.1-24.x.1.10, 312.96.1-24.x.1.11, 312.96.1-24.x.1.12, 312.96.1-24.x.1.13, 312.96.1-24.x.1.14, 312.96.1-24.x.1.15, 312.96.1-24.x.1.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 3 x y - z^{2} $
$=$ $6 x^{2} - 3 x y + 12 y^{2} - 5 z^{2} + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} + 6 x^{2} y^{2} - 9 x^{2} z^{2} + 9 z^{4} $
 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 embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{6048y^{2}z^{10}-9072y^{2}z^{8}w^{2}+432y^{2}z^{6}w^{4}+216y^{2}z^{4}w^{6}-1134y^{2}z^{2}w^{8}+189y^{2}w^{10}-992z^{12}+960z^{10}w^{2}+384z^{8}w^{4}-256z^{6}w^{6}+510z^{4}w^{8}-192z^{2}w^{10}+16w^{12}}{w^{4}z^{4}(6y^{2}z^{2}+3y^{2}w^{2}-2z^{4})}$

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
8.24.1.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.h.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.i.1 $24$ $2$ $2$ $0$ $0$ 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
24.96.1.e.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.p.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.ba.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bg.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bi.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bo.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bq.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.br.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.ed.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
24.192.9.cj.2 $24$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
120.96.1.dm.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.do.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dw.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ec.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.em.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.es.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.fa.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.fc.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.bv.2 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.rj.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.dm.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.do.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.dw.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ec.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.em.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.es.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fa.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fc.2 $168$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.dm.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.do.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.dw.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ec.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.em.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.es.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.fa.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.fc.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.dm.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.do.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.dw.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ec.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.em.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.es.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.fa.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.fc.2 $312$ $2$ $2$ $1$ $?$ dimension zero