↑

Properties

Label 24.144.8.er.1
Level $24$
Index $144$
Genus $8$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $144$
Index: $144$ $\PSL_2$-index:$144$
Genus: $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $2$ are rational) Cusp widths $6^{4}\cdot12^{2}\cdot24^{4}$ Cusp orbits $1^{2}\cdot2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24B8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.8.89

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&10\\4&7\end{bmatrix}$, $\begin{bmatrix}7&4\\4&17\end{bmatrix}$, $\begin{bmatrix}11&2\\20&1\end{bmatrix}$, $\begin{bmatrix}11&10\\16&9\end{bmatrix}$, $\begin{bmatrix}13&10\\4&11\end{bmatrix}$, $\begin{bmatrix}13&14\\20&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.8-24.er.1.1, 24.288.8-24.er.1.2, 24.288.8-24.er.1.3, 24.288.8-24.er.1.4, 24.288.8-24.er.1.5, 24.288.8-24.er.1.6, 24.288.8-24.er.1.7, 24.288.8-24.er.1.8, 24.288.8-24.er.1.9, 24.288.8-24.er.1.10, 24.288.8-24.er.1.11, 24.288.8-24.er.1.12, 24.288.8-24.er.1.13, 24.288.8-24.er.1.14, 24.288.8-24.er.1.15, 24.288.8-24.er.1.16, 24.288.8-24.er.1.17, 24.288.8-24.er.1.18, 24.288.8-24.er.1.19, 24.288.8-24.er.1.20, 24.288.8-24.er.1.21, 24.288.8-24.er.1.22, 24.288.8-24.er.1.23, 24.288.8-24.er.1.24, 24.288.8-24.er.1.25, 24.288.8-24.er.1.26, 24.288.8-24.er.1.27, 24.288.8-24.er.1.28, 24.288.8-24.er.1.29, 24.288.8-24.er.1.30, 24.288.8-24.er.1.31, 24.288.8-24.er.1.32, 120.288.8-24.er.1.1, 120.288.8-24.er.1.2, 120.288.8-24.er.1.3, 120.288.8-24.er.1.4, 120.288.8-24.er.1.5, 120.288.8-24.er.1.6, 120.288.8-24.er.1.7, 120.288.8-24.er.1.8, 120.288.8-24.er.1.9, 120.288.8-24.er.1.10, 120.288.8-24.er.1.11, 120.288.8-24.er.1.12, 120.288.8-24.er.1.13, 120.288.8-24.er.1.14, 120.288.8-24.er.1.15, 120.288.8-24.er.1.16, 120.288.8-24.er.1.17, 120.288.8-24.er.1.18, 120.288.8-24.er.1.19, 120.288.8-24.er.1.20, 120.288.8-24.er.1.21, 120.288.8-24.er.1.22, 120.288.8-24.er.1.23, 120.288.8-24.er.1.24, 120.288.8-24.er.1.25, 120.288.8-24.er.1.26, 120.288.8-24.er.1.27, 120.288.8-24.er.1.28, 120.288.8-24.er.1.29, 120.288.8-24.er.1.30, 120.288.8-24.er.1.31, 120.288.8-24.er.1.32, 168.288.8-24.er.1.1, 168.288.8-24.er.1.2, 168.288.8-24.er.1.3, 168.288.8-24.er.1.4, 168.288.8-24.er.1.5, 168.288.8-24.er.1.6, 168.288.8-24.er.1.7, 168.288.8-24.er.1.8, 168.288.8-24.er.1.9, 168.288.8-24.er.1.10, 168.288.8-24.er.1.11, 168.288.8-24.er.1.12, 168.288.8-24.er.1.13, 168.288.8-24.er.1.14, 168.288.8-24.er.1.15, 168.288.8-24.er.1.16, 168.288.8-24.er.1.17, 168.288.8-24.er.1.18, 168.288.8-24.er.1.19, 168.288.8-24.er.1.20, 168.288.8-24.er.1.21, 168.288.8-24.er.1.22, 168.288.8-24.er.1.23, 168.288.8-24.er.1.24, 168.288.8-24.er.1.25, 168.288.8-24.er.1.26, 168.288.8-24.er.1.27, 168.288.8-24.er.1.28, 168.288.8-24.er.1.29, 168.288.8-24.er.1.30, 168.288.8-24.er.1.31, 168.288.8-24.er.1.32, 264.288.8-24.er.1.1, 264.288.8-24.er.1.2, 264.288.8-24.er.1.3, 264.288.8-24.er.1.4, 264.288.8-24.er.1.5, 264.288.8-24.er.1.6, 264.288.8-24.er.1.7, 264.288.8-24.er.1.8, 264.288.8-24.er.1.9, 264.288.8-24.er.1.10, 264.288.8-24.er.1.11, 264.288.8-24.er.1.12, 264.288.8-24.er.1.13, 264.288.8-24.er.1.14, 264.288.8-24.er.1.15, 264.288.8-24.er.1.16, 264.288.8-24.er.1.17, 264.288.8-24.er.1.18, 264.288.8-24.er.1.19, 264.288.8-24.er.1.20, 264.288.8-24.er.1.21, 264.288.8-24.er.1.22, 264.288.8-24.er.1.23, 264.288.8-24.er.1.24, 264.288.8-24.er.1.25, 264.288.8-24.er.1.26, 264.288.8-24.er.1.27, 264.288.8-24.er.1.28, 264.288.8-24.er.1.29, 264.288.8-24.er.1.30, 264.288.8-24.er.1.31, 264.288.8-24.er.1.32, 312.288.8-24.er.1.1, 312.288.8-24.er.1.2, 312.288.8-24.er.1.3, 312.288.8-24.er.1.4, 312.288.8-24.er.1.5, 312.288.8-24.er.1.6, 312.288.8-24.er.1.7, 312.288.8-24.er.1.8, 312.288.8-24.er.1.9, 312.288.8-24.er.1.10, 312.288.8-24.er.1.11, 312.288.8-24.er.1.12, 312.288.8-24.er.1.13, 312.288.8-24.er.1.14, 312.288.8-24.er.1.15, 312.288.8-24.er.1.16, 312.288.8-24.er.1.17, 312.288.8-24.er.1.18, 312.288.8-24.er.1.19, 312.288.8-24.er.1.20, 312.288.8-24.er.1.21, 312.288.8-24.er.1.22, 312.288.8-24.er.1.23, 312.288.8-24.er.1.24, 312.288.8-24.er.1.25, 312.288.8-24.er.1.26, 312.288.8-24.er.1.27, 312.288.8-24.er.1.28, 312.288.8-24.er.1.29, 312.288.8-24.er.1.30, 312.288.8-24.er.1.31, 312.288.8-24.er.1.32
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{22}\cdot3^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{2}$
Newforms: 36.2.a.a$^{3}$, 72.2.d.a$^{2}$, 144.2.a.a

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 20 equations

$ 0 $ $=$ $ t^{2} - t u + u^{2} + v r $
$=$ $3 w^{2} + v r$
$=$ $2 w t - w u - w r - u r - v r$
$=$ $w v + w r - u v + u r + 2 v r$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ - 3 x^{8} + 12 x^{7} y - 3 x^{6} y^{2} - 60 x^{5} y^{3} + 7 x^{5} z^{3} + 159 x^{4} y^{4} + \cdots + 24 y^{8} $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:0:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x-y$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle 3w$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.72.4.y.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle x-y-2z$
$\displaystyle W$ $=$ $\displaystyle v+r$

Equation of the image curve:

$0$ $=$ $ 6XY-ZW $
$=$ $ 3X^{3}-24Y^{3}+XZ^{2}-YW^{2} $

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
24.48.0.w.2 $24$ $3$ $3$ $0$ $0$ full Jacobian
24.72.4.y.2 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
24.72.4.z.1 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
24.72.4.cd.1 $24$ $2$ $2$ $4$ $0$ $2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.288.15.ib.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.if.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.ix.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.jb.2 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.jn.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.jr.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.kd.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.kh.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.17.or.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.os.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.pf.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.ph.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.tc.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.td.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.te.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.tf.2 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bqe.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bqf.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.bqg.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bqh.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.bqu.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.bqv.1 $24$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
24.288.17.bqw.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.bqx.2 $24$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
120.288.15.dgu.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dhc.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dia.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dii.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.djg.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.djo.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dkm.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dku.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.17.jsk.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jsl.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jsm.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jsn.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jtq.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jtr.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jts.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jtt.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lug.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.luh.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lui.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.luj.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lvm.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lvn.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lvo.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lvp.1 $120$ $2$ $2$ $17$ $?$ not computed
168.288.15.ctv.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cud.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cvb.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cvj.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cwh.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cwp.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cxn.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cxv.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.17.jai.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jaj.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jak.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jal.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jbo.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jbp.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jbq.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jbr.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kxg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kxh.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kxi.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kxj.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kym.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kyn.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kyo.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kyp.1 $168$ $2$ $2$ $17$ $?$ not computed
264.288.15.ctv.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cud.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cvb.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cvj.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cwh.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cwp.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cxn.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cxv.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.17.jai.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jaj.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jak.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jal.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jbo.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jbp.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jbq.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jbr.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kya.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kyb.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kyc.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kyd.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kzg.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kzh.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kzi.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.kzj.1 $264$ $2$ $2$ $17$ $?$ not computed
312.288.15.cwn.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.cwv.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.cxt.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.cyb.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.cyz.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.czh.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.daf.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dan.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.17.jai.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jaj.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jak.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jal.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jbo.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jbp.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jbq.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jbr.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kxg.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kxh.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kxi.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kxj.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kym.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kyn.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kyo.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kyp.1 $312$ $2$ $2$ $17$ $?$ not computed