Modular curve 70.6720.257-70.ck.1.6

• Label: 70.6720.257-70.ck.1.6
• Level: $70$
• Index: $6720$
• Genus: $257$
• Analytic rank: $52$
• Cusps: $48$
• $\Q$-cusps: $2$

Invariants

Level:	$70$		$\SL_2$-level:	$70$		Newform level:	$4900$
Index:	$6720$		$\PSL_2$-index:	$3360$
Genus:	$257 = 1 + \frac{ 3360 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$
Cusps:	$48$ (of which $2$ are rational)		Cusp widths	$70^{48}$		Cusp orbits	$1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$52$
$\Q$-gonality:	$34 \le \gamma \le 56$
$\overline{\Q}$-gonality:	$34 \le \gamma \le 56$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label:

70.6720.257.18

Level structure

$\GL_2(\Z/70\Z)$-generators:	$\begin{bmatrix}4&19\\65&31\end{bmatrix}$, $\begin{bmatrix}11&50\\65&31\end{bmatrix}$, $\begin{bmatrix}57&58\\25&41\end{bmatrix}$
$\GL_2(\Z/70\Z)$-subgroup:	$C_3\times C_{12}:D_{12}$
Contains $-I$:	no $\quad$ (see 70.3360.257.ck.1 for the level structure with $-I$)
Cyclic 70-isogeny field degree:	$6$
Cyclic 70-torsion field degree:	$36$
Full 70-torsion field degree:	$864$

Jacobian

Conductor:	$2^{280}\cdot5^{410}\cdot7^{485}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{37}\cdot2^{44}\cdot3^{4}\cdot4^{23}\cdot6^{2}\cdot8^{2}$
Newforms:	35.2.a.a$^{2}$, 35.2.a.b$^{2}$, 35.2.b.a$^{2}$, 175.2.a.a, 175.2.a.b, 175.2.a.c, 175.2.a.d, 175.2.a.e, 175.2.a.f, 175.2.b.a, 175.2.b.b, 175.2.b.c, 196.2.a.a$^{3}$, 196.2.a.c$^{3}$, 245.2.a.e$^{2}$, 245.2.a.f$^{2}$, 245.2.a.h$^{2}$, 245.2.b.c$^{2}$, 245.2.b.e$^{2}$, 245.2.b.f$^{2}$, 980.2.a.b$^{2}$, 980.2.a.c$^{2}$, 980.2.a.e$^{2}$, 980.2.a.h$^{4}$, 980.2.a.i$^{2}$, 980.2.a.j$^{2}$, 980.2.a.k$^{2}$, 980.2.e.a$^{2}$, 980.2.e.b$^{2}$, 980.2.e.c$^{2}$, 980.2.e.e$^{2}$, 1225.2.a.ba, 1225.2.a.bb, 1225.2.a.bc, 1225.2.a.d, 1225.2.a.f, 1225.2.a.k, 1225.2.a.l, 1225.2.a.p, 1225.2.a.r, 1225.2.a.v, 1225.2.a.x, 1225.2.a.y, 1225.2.b.g, 1225.2.b.i, 1225.2.b.j, 1225.2.b.l, 1225.2.b.n, 4900.2.a.a, 4900.2.a.b, 4900.2.a.bb, 4900.2.a.bd, 4900.2.a.bf, 4900.2.a.bg, 4900.2.a.bh, 4900.2.a.bi, 4900.2.a.e$^{2}$, 4900.2.a.f, 4900.2.a.j, 4900.2.a.k, 4900.2.a.l, 4900.2.a.m, 4900.2.a.n, 4900.2.a.p, 4900.2.a.q, 4900.2.a.t, 4900.2.a.u, 4900.2.a.w, 4900.2.a.x, 4900.2.a.y, 4900.2.a.z, 4900.2.e.a, 4900.2.e.b, 4900.2.e.f$^{2}$, 4900.2.e.g, 4900.2.e.h, 4900.2.e.l, 4900.2.e.n, 4900.2.e.o, 4900.2.e.p, 4900.2.e.q, 4900.2.e.r, 4900.2.e.s, 4900.2.e.u

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points.

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{arith}}(5)$	$5$	$56$	$56$	$0$	$0$	full Jacobian
14.56.3.b.1	$14$	$120$	$60$	$3$	$1$	$1^{36}\cdot2^{43}\cdot3^{4}\cdot4^{23}\cdot6^{2}\cdot8^{2}$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
35.3360.117-35.a.1.7	$35$	$2$	$2$	$117$	$19$	$1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$
70.240.5-70.j.1.4	$70$	$28$	$28$	$5$	$1$	$1^{34}\cdot2^{43}\cdot3^{4}\cdot4^{23}\cdot6^{2}\cdot8^{2}$
70.1344.49-70.q.1.2	$70$	$5$	$5$	$49$	$7$	$1^{28}\cdot2^{32}\cdot3^{4}\cdot4^{19}\cdot6^{2}\cdot8^{2}$
70.1344.49-70.q.2.2	$70$	$5$	$5$	$49$	$7$	$1^{28}\cdot2^{32}\cdot3^{4}\cdot4^{19}\cdot6^{2}\cdot8^{2}$
70.3360.117-35.a.1.4	$70$	$2$	$2$	$117$	$19$	$1^{30}\cdot2^{23}\cdot3^{2}\cdot4^{11}\cdot6\cdot8$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
70.13440.513-70.cm.1.6	$70$	$2$	$2$	$513$	$98$	$1^{76}\cdot2^{54}\cdot3^{4}\cdot4^{12}\cdot6^{2}$
70.13440.513-70.cn.1.4	$70$	$2$	$2$	$513$	$109$	$1^{76}\cdot2^{54}\cdot3^{4}\cdot4^{12}\cdot6^{2}$
70.13440.513-70.co.1.4	$70$	$2$	$2$	$513$	$101$	$1^{76}\cdot2^{54}\cdot3^{4}\cdot4^{12}\cdot6^{2}$
70.13440.513-70.cp.1.6	$70$	$2$	$2$	$513$	$102$	$1^{76}\cdot2^{54}\cdot3^{4}\cdot4^{12}\cdot6^{2}$
70.20160.769-70.en.1.6	$70$	$3$	$3$	$769$	$141$	$1^{84}\cdot2^{102}\cdot3^{4}\cdot4^{42}\cdot6^{2}\cdot8^{4}$
70.20160.769-70.ep.1.6	$70$	$3$	$3$	$769$	$154$	$1^{100}\cdot2^{88}\cdot3^{12}\cdot4^{37}\cdot6^{6}\cdot8^{2}$