↑

Properties

Label 24.48.1.m.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: $32$
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: 24.48.1.316

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}13&2\\20&9\end{bmatrix}$, $\begin{bmatrix}13&4\\12&7\end{bmatrix}$, $\begin{bmatrix}15&22\\16&17\end{bmatrix}$, $\begin{bmatrix}19&18\\0&13\end{bmatrix}$, $\begin{bmatrix}23&4\\12&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.m.1.1, 24.96.1-24.m.1.2, 24.96.1-24.m.1.3, 24.96.1-24.m.1.4, 24.96.1-24.m.1.5, 24.96.1-24.m.1.6, 24.96.1-24.m.1.7, 24.96.1-24.m.1.8, 24.96.1-24.m.1.9, 24.96.1-24.m.1.10, 24.96.1-24.m.1.11, 24.96.1-24.m.1.12, 48.96.1-24.m.1.1, 48.96.1-24.m.1.2, 48.96.1-24.m.1.3, 48.96.1-24.m.1.4, 48.96.1-24.m.1.5, 48.96.1-24.m.1.6, 48.96.1-24.m.1.7, 48.96.1-24.m.1.8, 120.96.1-24.m.1.1, 120.96.1-24.m.1.2, 120.96.1-24.m.1.3, 120.96.1-24.m.1.4, 120.96.1-24.m.1.5, 120.96.1-24.m.1.6, 120.96.1-24.m.1.7, 120.96.1-24.m.1.8, 120.96.1-24.m.1.9, 120.96.1-24.m.1.10, 120.96.1-24.m.1.11, 120.96.1-24.m.1.12, 168.96.1-24.m.1.1, 168.96.1-24.m.1.2, 168.96.1-24.m.1.3, 168.96.1-24.m.1.4, 168.96.1-24.m.1.5, 168.96.1-24.m.1.6, 168.96.1-24.m.1.7, 168.96.1-24.m.1.8, 168.96.1-24.m.1.9, 168.96.1-24.m.1.10, 168.96.1-24.m.1.11, 168.96.1-24.m.1.12, 240.96.1-24.m.1.1, 240.96.1-24.m.1.2, 240.96.1-24.m.1.3, 240.96.1-24.m.1.4, 240.96.1-24.m.1.5, 240.96.1-24.m.1.6, 240.96.1-24.m.1.7, 240.96.1-24.m.1.8, 264.96.1-24.m.1.1, 264.96.1-24.m.1.2, 264.96.1-24.m.1.3, 264.96.1-24.m.1.4, 264.96.1-24.m.1.5, 264.96.1-24.m.1.6, 264.96.1-24.m.1.7, 264.96.1-24.m.1.8, 264.96.1-24.m.1.9, 264.96.1-24.m.1.10, 264.96.1-24.m.1.11, 264.96.1-24.m.1.12, 312.96.1-24.m.1.1, 312.96.1-24.m.1.2, 312.96.1-24.m.1.3, 312.96.1-24.m.1.4, 312.96.1-24.m.1.5, 312.96.1-24.m.1.6, 312.96.1-24.m.1.7, 312.96.1-24.m.1.8, 312.96.1-24.m.1.9, 312.96.1-24.m.1.10, 312.96.1-24.m.1.11, 312.96.1-24.m.1.12
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

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

Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} - 3 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{189y^{2}z^{10}+567y^{2}z^{8}w^{2}+54y^{2}z^{6}w^{4}-54y^{2}z^{4}w^{6}-567y^{2}z^{2}w^{8}-189y^{2}w^{10}-32z^{12}-192z^{10}w^{2}-255z^{8}w^{4}-64z^{6}w^{6}-48z^{4}w^{8}+60z^{2}w^{10}+31w^{12}}{w^{4}z^{4}(3y^{2}z^{2}-3y^{2}w^{2}+w^{4})}$

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.c.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.h.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.em.2 $24$ $2$ $2$ $0$ $0$ 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.f.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.i.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.j.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.k.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.co.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
24.192.9.bg.2 $24$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
48.96.5.q.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.r.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.s.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.t.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.u.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.v.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.w.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.x.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
120.96.1.i.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.j.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.k.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.l.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.y.2 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qa.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.i.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.j.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.k.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.l.2 $168$ $2$ $2$ $1$ $?$ dimension zero
240.96.5.bo.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bp.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.br.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bs.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bt.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bu.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bv.1 $240$ $2$ $2$ $5$ $?$ not computed
264.96.1.i.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.j.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.k.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.l.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.i.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.j.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.k.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.l.2 $312$ $2$ $2$ $1$ $?$ dimension zero