Properties

Label 24.24.1.dh.1
Level $24$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $2^{2}$
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: 8C1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.24.1.56

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}9&8\\2&11\end{bmatrix}$, $\begin{bmatrix}11&11\\10&17\end{bmatrix}$, $\begin{bmatrix}15&17\\14&1\end{bmatrix}$, $\begin{bmatrix}19&6\\8&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $3072$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

$ 0 $ $=$ $ 2 x y - z w $
$=$ $16 x^{2} - 6 y^{2} - 3 z^{2} + 4 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 6 x^{4} + 3 x^{2} y^{2} - x^{2} z^{2} - y^{2} z^{2} $
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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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Maps to other modular curves

$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^6\cdot3\,\frac{8730y^{2}z^{4}+6480y^{2}z^{2}w^{2}+1120y^{2}w^{4}-243z^{6}-9372z^{4}w^{2}-4496z^{2}w^{4}-576w^{6}}{54y^{2}z^{4}+144y^{2}z^{2}w^{2}-96y^{2}w^{4}+27z^{6}+108z^{4}w^{2}-48z^{2}w^{4}+64w^{6}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.t.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.bq.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.12.1.bz.1 $24$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.48.1.t.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.cy.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.eb.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ek.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.kb.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.kp.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.lb.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ll.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.5.kr.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
24.96.5.ej.1 $24$ $4$ $4$ $5$ $1$ $1^{4}$
120.48.1.bhn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bhr.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bid.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bih.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bsh.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bsl.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bsx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.btb.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.120.9.qn.1 $120$ $5$ $5$ $9$ $?$ not computed
120.144.9.oaf.1 $120$ $6$ $6$ $9$ $?$ not computed
120.240.17.ewp.1 $120$ $10$ $10$ $17$ $?$ not computed
168.48.1.bhl.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bhp.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bib.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bif.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bsf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bsj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bsv.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.bsz.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.192.13.kl.1 $168$ $8$ $8$ $13$ $?$ not computed
264.48.1.bhl.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bhp.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bib.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bif.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bsf.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bsj.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bsv.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.bsz.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.288.21.ip.1 $264$ $12$ $12$ $21$ $?$ not computed
312.48.1.bhn.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bhr.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bid.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bih.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bsh.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bsl.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.bsx.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.btb.1 $312$ $2$ $2$ $1$ $?$ dimension zero