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Properties

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

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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: $10$ (none of which are rational) Cusp widths $6^{4}\cdot12^{2}\cdot24^{4}$ Cusp orbits $2^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 6$
$\overline{\Q}$-gonality: $3$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24B8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.144.8.118

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&14\\20&3\end{bmatrix}$, $\begin{bmatrix}3&8\\20&17\end{bmatrix}$, $\begin{bmatrix}7&8\\8&11\end{bmatrix}$, $\begin{bmatrix}11&6\\12&17\end{bmatrix}$, $\begin{bmatrix}11&18\\12&5\end{bmatrix}$, $\begin{bmatrix}17&14\\12&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.8-24.fo.2.1, 24.288.8-24.fo.2.2, 24.288.8-24.fo.2.3, 24.288.8-24.fo.2.4, 24.288.8-24.fo.2.5, 24.288.8-24.fo.2.6, 24.288.8-24.fo.2.7, 24.288.8-24.fo.2.8, 24.288.8-24.fo.2.9, 24.288.8-24.fo.2.10, 24.288.8-24.fo.2.11, 24.288.8-24.fo.2.12, 24.288.8-24.fo.2.13, 24.288.8-24.fo.2.14, 24.288.8-24.fo.2.15, 24.288.8-24.fo.2.16, 24.288.8-24.fo.2.17, 24.288.8-24.fo.2.18, 24.288.8-24.fo.2.19, 24.288.8-24.fo.2.20, 24.288.8-24.fo.2.21, 24.288.8-24.fo.2.22, 24.288.8-24.fo.2.23, 24.288.8-24.fo.2.24, 48.288.8-24.fo.2.1, 48.288.8-24.fo.2.2, 48.288.8-24.fo.2.3, 48.288.8-24.fo.2.4, 48.288.8-24.fo.2.5, 48.288.8-24.fo.2.6, 48.288.8-24.fo.2.7, 48.288.8-24.fo.2.8, 48.288.8-24.fo.2.9, 48.288.8-24.fo.2.10, 48.288.8-24.fo.2.11, 48.288.8-24.fo.2.12, 48.288.8-24.fo.2.13, 48.288.8-24.fo.2.14, 48.288.8-24.fo.2.15, 48.288.8-24.fo.2.16, 120.288.8-24.fo.2.1, 120.288.8-24.fo.2.2, 120.288.8-24.fo.2.3, 120.288.8-24.fo.2.4, 120.288.8-24.fo.2.5, 120.288.8-24.fo.2.6, 120.288.8-24.fo.2.7, 120.288.8-24.fo.2.8, 120.288.8-24.fo.2.9, 120.288.8-24.fo.2.10, 120.288.8-24.fo.2.11, 120.288.8-24.fo.2.12, 120.288.8-24.fo.2.13, 120.288.8-24.fo.2.14, 120.288.8-24.fo.2.15, 120.288.8-24.fo.2.16, 120.288.8-24.fo.2.17, 120.288.8-24.fo.2.18, 120.288.8-24.fo.2.19, 120.288.8-24.fo.2.20, 120.288.8-24.fo.2.21, 120.288.8-24.fo.2.22, 120.288.8-24.fo.2.23, 120.288.8-24.fo.2.24, 168.288.8-24.fo.2.1, 168.288.8-24.fo.2.2, 168.288.8-24.fo.2.3, 168.288.8-24.fo.2.4, 168.288.8-24.fo.2.5, 168.288.8-24.fo.2.6, 168.288.8-24.fo.2.7, 168.288.8-24.fo.2.8, 168.288.8-24.fo.2.9, 168.288.8-24.fo.2.10, 168.288.8-24.fo.2.11, 168.288.8-24.fo.2.12, 168.288.8-24.fo.2.13, 168.288.8-24.fo.2.14, 168.288.8-24.fo.2.15, 168.288.8-24.fo.2.16, 168.288.8-24.fo.2.17, 168.288.8-24.fo.2.18, 168.288.8-24.fo.2.19, 168.288.8-24.fo.2.20, 168.288.8-24.fo.2.21, 168.288.8-24.fo.2.22, 168.288.8-24.fo.2.23, 168.288.8-24.fo.2.24, 240.288.8-24.fo.2.1, 240.288.8-24.fo.2.2, 240.288.8-24.fo.2.3, 240.288.8-24.fo.2.4, 240.288.8-24.fo.2.5, 240.288.8-24.fo.2.6, 240.288.8-24.fo.2.7, 240.288.8-24.fo.2.8, 240.288.8-24.fo.2.9, 240.288.8-24.fo.2.10, 240.288.8-24.fo.2.11, 240.288.8-24.fo.2.12, 240.288.8-24.fo.2.13, 240.288.8-24.fo.2.14, 240.288.8-24.fo.2.15, 240.288.8-24.fo.2.16, 264.288.8-24.fo.2.1, 264.288.8-24.fo.2.2, 264.288.8-24.fo.2.3, 264.288.8-24.fo.2.4, 264.288.8-24.fo.2.5, 264.288.8-24.fo.2.6, 264.288.8-24.fo.2.7, 264.288.8-24.fo.2.8, 264.288.8-24.fo.2.9, 264.288.8-24.fo.2.10, 264.288.8-24.fo.2.11, 264.288.8-24.fo.2.12, 264.288.8-24.fo.2.13, 264.288.8-24.fo.2.14, 264.288.8-24.fo.2.15, 264.288.8-24.fo.2.16, 264.288.8-24.fo.2.17, 264.288.8-24.fo.2.18, 264.288.8-24.fo.2.19, 264.288.8-24.fo.2.20, 264.288.8-24.fo.2.21, 264.288.8-24.fo.2.22, 264.288.8-24.fo.2.23, 264.288.8-24.fo.2.24, 312.288.8-24.fo.2.1, 312.288.8-24.fo.2.2, 312.288.8-24.fo.2.3, 312.288.8-24.fo.2.4, 312.288.8-24.fo.2.5, 312.288.8-24.fo.2.6, 312.288.8-24.fo.2.7, 312.288.8-24.fo.2.8, 312.288.8-24.fo.2.9, 312.288.8-24.fo.2.10, 312.288.8-24.fo.2.11, 312.288.8-24.fo.2.12, 312.288.8-24.fo.2.13, 312.288.8-24.fo.2.14, 312.288.8-24.fo.2.15, 312.288.8-24.fo.2.16, 312.288.8-24.fo.2.17, 312.288.8-24.fo.2.18, 312.288.8-24.fo.2.19, 312.288.8-24.fo.2.20, 312.288.8-24.fo.2.21, 312.288.8-24.fo.2.22, 312.288.8-24.fo.2.23, 312.288.8-24.fo.2.24
Cyclic 24-isogeny field degree: $4$
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 v - u^{2} - u r + v^{2} - r^{2} $
$=$ $y^{2} - t u - u^{2} - u r - r^{2}$
$=$ $x y + x u + y z + w r$
$=$ $x y + x t + x v + y z - w v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{10} - 4 x^{9} z + 10 x^{8} z^{2} - 16 x^{7} z^{3} + 19 x^{6} z^{4} - 16 x^{5} z^{5} + \cdots + 144 y^{6} z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle z$

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 :

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

Equation of the image curve:

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

Modular covers

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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.bc.2 $24$ $3$ $3$ $0$ $0$ full Jacobian
24.72.4.y.2 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
24.72.4.ch.1 $24$ $2$ $2$ $4$ $0$ $2^{2}$
24.72.4.gk.2 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$

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.ky.2 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.le.2 $24$ $2$ $2$ $15$ $2$ $1^{3}\cdot2^{2}$
24.288.15.lw.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.mc.2 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.nr.2 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.ny.1 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.oq.2 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.15.ow.2 $24$ $2$ $2$ $15$ $0$ $1^{3}\cdot2^{2}$
24.288.17.ol.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.tm.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.ble.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.blm.2 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bru.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.bsc.1 $24$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
24.288.17.btm.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.btu.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.v.2 $48$ $2$ $2$ $17$ $0$ $1^{5}\cdot2^{2}$
48.288.17.y.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.ci.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.cr.2 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.di.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.ed.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.17.ej.2 $48$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
48.288.17.ey.2 $48$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
48.288.19.jd.2 $48$ $2$ $2$ $19$ $2$ $1^{5}\cdot2\cdot4$
48.288.19.ka.2 $48$ $2$ $2$ $19$ $1$ $1^{5}\cdot2\cdot4$
48.288.19.ma.2 $48$ $2$ $2$ $19$ $1$ $1^{5}\cdot2\cdot4$
48.288.19.mz.2 $48$ $2$ $2$ $19$ $1$ $1^{5}\cdot2\cdot4$
48.288.19.oq.1 $48$ $2$ $2$ $19$ $1$ $1^{5}\cdot2\cdot4$
48.288.19.pa.2 $48$ $2$ $2$ $19$ $2$ $1^{5}\cdot2\cdot4$
48.288.19.qb.2 $48$ $2$ $2$ $19$ $0$ $1^{5}\cdot2\cdot4$
48.288.19.qf.2 $48$ $2$ $2$ $19$ $1$ $1^{5}\cdot2\cdot4$
120.288.15.dmd.2 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dmp.2 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dnz.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dol.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dpv.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dqh.2 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.drr.2 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dsd.2 $120$ $2$ $2$ $15$ $?$ not computed
120.288.17.jvo.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jwe.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jya.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jyq.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lxk.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lya.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lzw.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.mam.2 $120$ $2$ $2$ $17$ $?$ not computed
168.288.15.cze.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.czq.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dba.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dbm.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dcw.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.ddi.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.des.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dfe.2 $168$ $2$ $2$ $15$ $?$ not computed
168.288.17.jdm.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jec.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jfy.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jgo.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lak.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lba.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lcw.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.ldm.2 $168$ $2$ $2$ $17$ $?$ not computed
240.288.17.fh.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.fk.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ig.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ip.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.kg.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ln.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.mf.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.17.ng.2 $240$ $2$ $2$ $17$ $?$ not computed
240.288.19.btz.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.bvg.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.bwi.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.bxv.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.cds.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.ced.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.cgh.2 $240$ $2$ $2$ $19$ $?$ not computed
240.288.19.cgm.2 $240$ $2$ $2$ $19$ $?$ not computed
264.288.15.cze.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.czq.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dba.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dbm.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dcw.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.ddi.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.des.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dfe.2 $264$ $2$ $2$ $15$ $?$ not computed
264.288.17.jdm.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jec.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jfy.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jgo.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lbe.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lbu.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.ldq.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.leg.2 $264$ $2$ $2$ $17$ $?$ not computed
312.288.15.dbw.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dci.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dds.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dee.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dfo.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dga.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dhk.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dhw.2 $312$ $2$ $2$ $15$ $?$ not computed
312.288.17.jdm.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jec.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jfy.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jgo.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lak.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lba.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lcw.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.ldm.2 $312$ $2$ $2$ $17$ $?$ not computed