Modular curve 312.144.3-24.ua.1.4

• Label: 312.144.3-24.ua.1.4
• Level: $312$
• Index: $144$
• Genus: $3$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$312$		$\SL_2$-level:	$12$		Newform level:	$576$
Index:	$144$		$\PSL_2$-index:	$72$
Genus:	$3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$6^{4}\cdot12^{4}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 3$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 3$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12D3

Level structure

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}91&259\\108&113\end{bmatrix}$, $\begin{bmatrix}145&304\\54&119\end{bmatrix}$, $\begin{bmatrix}187&106\\288&275\end{bmatrix}$, $\begin{bmatrix}253&66\\174&109\end{bmatrix}$, $\begin{bmatrix}263&191\\90&289\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 24.72.3.ua.1 for the level structure with $-I$)
Cyclic 312-isogeny field degree:	$56$
Cyclic 312-torsion field degree:	$5376$
Full 312-torsion field degree:	$13418496$

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $	$=$	$ x^{2} t - 2 x z t - x w t - y^{2} t - y z t + z^{2} t + z w t $
	$=$	$2 x^{2} t + 3 x y t + y^{2} t + y w t + z w t$
	$=$	$2 x z t + x w t + y z t + 2 y w t + 2 z w t + w^{2} t$
	$=$	$2 x^{2} w + 3 x y w + y^{2} w + y w^{2} + z w^{2}$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 52 x^{6} + 138 x^{5} z - 48 x^{4} y^{2} + 141 x^{4} z^{2} - 120 x^{3} y^{2} z + 80 x^{3} z^{3} + \cdots + z^{6} $

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ -2x^{7} + 14x^{4} + 16x $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model

$(0:0:0:0:1)$, $(-1:2:-2:1:0)$, $(-1:1:0:0:0)$, $(-1:0:-2:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{2\cdot3\cdot13^2}\cdot\frac{12875780008320xw^{10}-26542469881056xw^{8}t^{2}+17027176029180xw^{6}t^{4}-1646249466168xw^{4}t^{6}+1144990119780xw^{2}t^{8}-20494830096xt^{10}+1318832736y^{11}-8269437696y^{9}t^{2}+28158861120y^{7}t^{4}-45477451812y^{5}t^{6}+35459224086y^{3}t^{8}-16483251783168yzw^{9}-3228832129968yzw^{7}t^{2}-13943991245268yzw^{5}t^{4}-1412417490588yzw^{3}t^{6}-132521313834yzwt^{8}+16198440219648yw^{10}-21020064425376yw^{8}t^{2}+23837855985720yw^{6}t^{4}-17555485176yw^{4}t^{6}+1256984820702yw^{2}t^{8}-38153153883yt^{10}-10902622337056z^{2}w^{9}+24615318650288z^{2}w^{7}t^{2}-920214327704z^{2}w^{5}t^{4}+3379702812564z^{2}w^{3}t^{6}-48662061786z^{2}wt^{8}+8139756791552zw^{10}+25202867950928zw^{8}t^{2}+20657748000592zw^{6}t^{4}+3600328832436zw^{4}t^{6}-34650943470zw^{2}t^{8}-23180391351zt^{10}+5752541075456w^{11}+22427050652720w^{9}t^{2}+5104528487596w^{7}t^{4}+2027231164416w^{5}t^{6}-586373755500w^{3}t^{8}-3523198770wt^{10}}{t^{6}(400120xw^{4}-196405xw^{2}t^{2}+125892xt^{4}-487972yzw^{3}-128934yzwt^{2}+544380yw^{4}-135812yw^{2}t^{2}+125892yt^{4}-280078z^{2}w^{3}+152308z^{2}wt^{2}+352154zw^{4}+145733zw^{2}t^{2}-37648zt^{4}+207894w^{5}+161228w^{3}t^{2}-81770wt^{4})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.72.3.ua.1 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{2}t$
$\displaystyle Z$	$=$	$\displaystyle 2w$

Equation of the image curve:

$0$

$=$

$ 52X^{6}-48X^{4}Y^{2}-288X^{2}Y^{4}+138X^{5}Z-120X^{3}Y^{2}Z+72XY^{4}Z+141X^{4}Z^{2}-108X^{2}Y^{2}Z^{2}+36Y^{4}Z^{2}+80X^{3}Z^{3}-48XY^{2}Z^{3}+30X^{2}Z^{4}-12Y^{2}Z^{4}+6XZ^{5}+Z^{6} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.72.3.ua.1 :

$\displaystyle X$	$=$	$\displaystyle \frac{1}{6}z^{4}-\frac{1}{6}z^{3}w-\frac{5}{4}z^{2}w^{2}-\frac{1}{4}z^{2}t^{2}-\frac{2}{3}zw^{3}+\frac{1}{8}zwt^{2}-\frac{1}{3}w^{4}+\frac{1}{8}w^{2}t^{2}$
$\displaystyle Y$	$=$	$\displaystyle -\frac{3}{16}z^{15}t-\frac{165}{64}z^{14}wt-\frac{939}{64}z^{13}w^{2}t-\frac{9}{64}z^{13}t^{3}-\frac{11793}{256}z^{12}w^{3}t-\frac{135}{128}z^{12}wt^{3}-\frac{23007}{256}z^{11}w^{4}t-\frac{837}{256}z^{11}w^{2}t^{3}-\frac{14901}{128}z^{10}w^{5}t-\frac{2637}{512}z^{10}w^{3}t^{3}-\frac{3291}{32}z^{9}w^{6}t-\frac{1431}{512}z^{9}w^{4}t^{3}-\frac{483}{8}z^{8}w^{7}t+\frac{999}{256}z^{8}w^{5}t^{3}-\frac{363}{16}z^{7}w^{8}t+\frac{819}{128}z^{7}w^{6}t^{3}-\frac{45}{8}z^{6}w^{9}t+\frac{135}{64}z^{6}w^{7}t^{3}$
$\displaystyle Z$	$=$	$\displaystyle -\frac{1}{3}z^{4}-\frac{17}{12}z^{3}w-\frac{7}{4}z^{2}w^{2}-\frac{1}{4}z^{2}t^{2}-\frac{2}{3}zw^{3}+\frac{1}{8}zwt^{2}-\frac{1}{3}w^{4}+\frac{1}{8}w^{2}t^{2}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
78.72.0-6.a.1.2	$78$	$2$	$2$	$0$	$?$
312.48.1-24.es.1.9	$312$	$3$	$3$	$1$	$?$
312.48.1-24.es.1.13	$312$	$3$	$3$	$1$	$?$
312.72.0-6.a.1.4	$312$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
312.288.5-24.d.1.2	$312$	$2$	$2$	$5$
312.288.5-24.ba.1.4	$312$	$2$	$2$	$5$
312.288.5-24.cf.1.3	$312$	$2$	$2$	$5$
312.288.5-24.ch.1.6	$312$	$2$	$2$	$5$
312.288.5-24.hs.1.4	$312$	$2$	$2$	$5$
312.288.5-24.ht.1.3	$312$	$2$	$2$	$5$
312.288.5-24.hz.1.4	$312$	$2$	$2$	$5$
312.288.5-24.ib.1.1	$312$	$2$	$2$	$5$
312.288.5-312.boi.1.1	$312$	$2$	$2$	$5$
312.288.5-312.boj.1.8	$312$	$2$	$2$	$5$
312.288.5-312.bop.1.8	$312$	$2$	$2$	$5$
312.288.5-312.boq.1.7	$312$	$2$	$2$	$5$
312.288.5-312.bzc.1.6	$312$	$2$	$2$	$5$
312.288.5-312.bzd.1.8	$312$	$2$	$2$	$5$
312.288.5-312.bzj.1.8	$312$	$2$	$2$	$5$
312.288.5-312.bzk.1.1	$312$	$2$	$2$	$5$