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Properties

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

Related objects

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $576$
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 $1^{2}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-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: 24.144.8.9

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&8\\20&23\end{bmatrix}$, $\begin{bmatrix}11&10\\4&13\end{bmatrix}$, $\begin{bmatrix}11&20\\16&17\end{bmatrix}$, $\begin{bmatrix}15&8\\20&9\end{bmatrix}$, $\begin{bmatrix}19&16\\4&23\end{bmatrix}$, $\begin{bmatrix}23&4\\20&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.288.8-24.fd.1.1, 24.288.8-24.fd.1.2, 24.288.8-24.fd.1.3, 24.288.8-24.fd.1.4, 24.288.8-24.fd.1.5, 24.288.8-24.fd.1.6, 24.288.8-24.fd.1.7, 24.288.8-24.fd.1.8, 24.288.8-24.fd.1.9, 24.288.8-24.fd.1.10, 24.288.8-24.fd.1.11, 24.288.8-24.fd.1.12, 24.288.8-24.fd.1.13, 24.288.8-24.fd.1.14, 24.288.8-24.fd.1.15, 24.288.8-24.fd.1.16, 24.288.8-24.fd.1.17, 24.288.8-24.fd.1.18, 24.288.8-24.fd.1.19, 24.288.8-24.fd.1.20, 24.288.8-24.fd.1.21, 24.288.8-24.fd.1.22, 24.288.8-24.fd.1.23, 24.288.8-24.fd.1.24, 24.288.8-24.fd.1.25, 24.288.8-24.fd.1.26, 24.288.8-24.fd.1.27, 24.288.8-24.fd.1.28, 24.288.8-24.fd.1.29, 24.288.8-24.fd.1.30, 24.288.8-24.fd.1.31, 24.288.8-24.fd.1.32, 120.288.8-24.fd.1.1, 120.288.8-24.fd.1.2, 120.288.8-24.fd.1.3, 120.288.8-24.fd.1.4, 120.288.8-24.fd.1.5, 120.288.8-24.fd.1.6, 120.288.8-24.fd.1.7, 120.288.8-24.fd.1.8, 120.288.8-24.fd.1.9, 120.288.8-24.fd.1.10, 120.288.8-24.fd.1.11, 120.288.8-24.fd.1.12, 120.288.8-24.fd.1.13, 120.288.8-24.fd.1.14, 120.288.8-24.fd.1.15, 120.288.8-24.fd.1.16, 120.288.8-24.fd.1.17, 120.288.8-24.fd.1.18, 120.288.8-24.fd.1.19, 120.288.8-24.fd.1.20, 120.288.8-24.fd.1.21, 120.288.8-24.fd.1.22, 120.288.8-24.fd.1.23, 120.288.8-24.fd.1.24, 120.288.8-24.fd.1.25, 120.288.8-24.fd.1.26, 120.288.8-24.fd.1.27, 120.288.8-24.fd.1.28, 120.288.8-24.fd.1.29, 120.288.8-24.fd.1.30, 120.288.8-24.fd.1.31, 120.288.8-24.fd.1.32, 168.288.8-24.fd.1.1, 168.288.8-24.fd.1.2, 168.288.8-24.fd.1.3, 168.288.8-24.fd.1.4, 168.288.8-24.fd.1.5, 168.288.8-24.fd.1.6, 168.288.8-24.fd.1.7, 168.288.8-24.fd.1.8, 168.288.8-24.fd.1.9, 168.288.8-24.fd.1.10, 168.288.8-24.fd.1.11, 168.288.8-24.fd.1.12, 168.288.8-24.fd.1.13, 168.288.8-24.fd.1.14, 168.288.8-24.fd.1.15, 168.288.8-24.fd.1.16, 168.288.8-24.fd.1.17, 168.288.8-24.fd.1.18, 168.288.8-24.fd.1.19, 168.288.8-24.fd.1.20, 168.288.8-24.fd.1.21, 168.288.8-24.fd.1.22, 168.288.8-24.fd.1.23, 168.288.8-24.fd.1.24, 168.288.8-24.fd.1.25, 168.288.8-24.fd.1.26, 168.288.8-24.fd.1.27, 168.288.8-24.fd.1.28, 168.288.8-24.fd.1.29, 168.288.8-24.fd.1.30, 168.288.8-24.fd.1.31, 168.288.8-24.fd.1.32, 264.288.8-24.fd.1.1, 264.288.8-24.fd.1.2, 264.288.8-24.fd.1.3, 264.288.8-24.fd.1.4, 264.288.8-24.fd.1.5, 264.288.8-24.fd.1.6, 264.288.8-24.fd.1.7, 264.288.8-24.fd.1.8, 264.288.8-24.fd.1.9, 264.288.8-24.fd.1.10, 264.288.8-24.fd.1.11, 264.288.8-24.fd.1.12, 264.288.8-24.fd.1.13, 264.288.8-24.fd.1.14, 264.288.8-24.fd.1.15, 264.288.8-24.fd.1.16, 264.288.8-24.fd.1.17, 264.288.8-24.fd.1.18, 264.288.8-24.fd.1.19, 264.288.8-24.fd.1.20, 264.288.8-24.fd.1.21, 264.288.8-24.fd.1.22, 264.288.8-24.fd.1.23, 264.288.8-24.fd.1.24, 264.288.8-24.fd.1.25, 264.288.8-24.fd.1.26, 264.288.8-24.fd.1.27, 264.288.8-24.fd.1.28, 264.288.8-24.fd.1.29, 264.288.8-24.fd.1.30, 264.288.8-24.fd.1.31, 264.288.8-24.fd.1.32, 312.288.8-24.fd.1.1, 312.288.8-24.fd.1.2, 312.288.8-24.fd.1.3, 312.288.8-24.fd.1.4, 312.288.8-24.fd.1.5, 312.288.8-24.fd.1.6, 312.288.8-24.fd.1.7, 312.288.8-24.fd.1.8, 312.288.8-24.fd.1.9, 312.288.8-24.fd.1.10, 312.288.8-24.fd.1.11, 312.288.8-24.fd.1.12, 312.288.8-24.fd.1.13, 312.288.8-24.fd.1.14, 312.288.8-24.fd.1.15, 312.288.8-24.fd.1.16, 312.288.8-24.fd.1.17, 312.288.8-24.fd.1.18, 312.288.8-24.fd.1.19, 312.288.8-24.fd.1.20, 312.288.8-24.fd.1.21, 312.288.8-24.fd.1.22, 312.288.8-24.fd.1.23, 312.288.8-24.fd.1.24, 312.288.8-24.fd.1.25, 312.288.8-24.fd.1.26, 312.288.8-24.fd.1.27, 312.288.8-24.fd.1.28, 312.288.8-24.fd.1.29, 312.288.8-24.fd.1.30, 312.288.8-24.fd.1.31, 312.288.8-24.fd.1.32
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $512$

Jacobian

Conductor: $2^{28}\cdot3^{16}$
Simple: no
Squarefree: no
Decomposition: $1^{4}\cdot2^{2}$
Newforms: 36.2.a.a$^{2}$, 72.2.d.a$^{2}$, 576.2.a.e, 576.2.a.f

Models

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

$ 0 $ $=$ $ x r + w u $
$=$ $x t - y t + z w$
$=$ $x v + y v - z w$
$=$ $x t - y t + y r - z w$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 16 x^{6} z^{2} - 4 x^{5} y^{3} - 8 x^{4} z^{4} + 12 x^{3} y^{3} z^{2} - 4 x^{2} z^{6} - x y^{3} z^{4} + 2 z^{8} $
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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
$(0:0:0:0:1:0:0:0)$, $(0:0:0:0:0:0:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+y$
$\displaystyle Y$ $=$ $\displaystyle 4z$
$\displaystyle Z$ $=$ $\displaystyle w$

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.z.1 :

$\displaystyle X$ $=$ $\displaystyle -x$
$\displaystyle Y$ $=$ $\displaystyle -y$
$\displaystyle Z$ $=$ $\displaystyle z$
$\displaystyle W$ $=$ $\displaystyle -u$

Equation of the image curve:

$0$ $=$ $ 2XZ-YW $
$=$ $ X^{3}-XY^{2}+8Z^{3}-ZW^{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

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}^+(3)$ $3$ $48$ $48$ $0$ $0$ full Jacobian
8.48.0.i.1 $8$ $3$ $3$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.i.1 $8$ $3$ $3$ $0$ $0$ full Jacobian
24.72.4.z.1 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
24.72.4.z.2 $24$ $2$ $2$ $4$ $0$ $1^{2}\cdot2$
24.72.4.cg.1 $24$ $2$ $2$ $4$ $1$ $2^{2}$

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.kp.1 $24$ $2$ $2$ $15$ $2$ $1^{3}\cdot2^{2}$
24.288.15.kt.1 $24$ $2$ $2$ $15$ $2$ $1^{3}\cdot2^{2}$
24.288.15.ln.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.lr.1 $24$ $2$ $2$ $15$ $2$ $1^{3}\cdot2^{2}$
24.288.15.ni.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.nk.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.oj.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.15.ol.1 $24$ $2$ $2$ $15$ $1$ $1^{3}\cdot2^{2}$
24.288.17.oz.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.oz.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.pn.1 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.pn.2 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.bkw.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bkw.2 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bkx.1 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.bkx.2 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.brm.1 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.brm.2 $24$ $2$ $2$ $17$ $1$ $1^{5}\cdot2^{2}$
24.288.17.brn.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.brn.2 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.bte.1 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.bte.2 $24$ $2$ $2$ $17$ $3$ $1^{5}\cdot2^{2}$
24.288.17.btf.1 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
24.288.17.btf.2 $24$ $2$ $2$ $17$ $2$ $1^{5}\cdot2^{2}$
120.288.15.dlo.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dls.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dnk.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dno.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dpg.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.dpk.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.drc.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.15.drg.1 $120$ $2$ $2$ $15$ $?$ not computed
120.288.17.juy.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.juy.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.juz.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.juz.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jxk.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jxk.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jxl.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.jxl.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lwu.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lwu.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lwv.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lwv.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lzg.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lzg.2 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lzh.1 $120$ $2$ $2$ $17$ $?$ not computed
120.288.17.lzh.2 $120$ $2$ $2$ $17$ $?$ not computed
168.288.15.cyp.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.cyt.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dal.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dap.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dch.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.dcl.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.ded.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.15.deh.1 $168$ $2$ $2$ $15$ $?$ not computed
168.288.17.jcw.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jcw.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jcx.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jcx.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jfi.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jfi.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jfj.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.jfj.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kzu.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kzu.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kzv.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.kzv.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lcg.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lcg.2 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lch.1 $168$ $2$ $2$ $17$ $?$ not computed
168.288.17.lch.2 $168$ $2$ $2$ $17$ $?$ not computed
264.288.15.cyp.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.cyt.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dal.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dap.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dch.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.dcl.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.ded.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.15.deh.1 $264$ $2$ $2$ $15$ $?$ not computed
264.288.17.jcw.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jcw.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jcx.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jcx.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jfi.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jfi.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jfj.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.jfj.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lao.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lao.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lap.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lap.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lda.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.lda.2 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.ldb.1 $264$ $2$ $2$ $17$ $?$ not computed
264.288.17.ldb.2 $264$ $2$ $2$ $17$ $?$ not computed
312.288.15.dbh.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dbl.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.ddd.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.ddh.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dez.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dfd.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dgv.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.15.dgz.1 $312$ $2$ $2$ $15$ $?$ not computed
312.288.17.jcw.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jcw.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jcx.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jcx.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jfi.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jfi.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jfj.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.jfj.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kzu.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kzu.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kzv.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.kzv.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lcg.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lcg.2 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lch.1 $312$ $2$ $2$ $17$ $?$ not computed
312.288.17.lch.2 $312$ $2$ $2$ $17$ $?$ not computed