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Properties

Label 56.2016.70-56.l.1.38
Level $56$
Index $2016$
Genus $70$
Analytic rank $6$
Cusps $30$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $784$
Index: $2016$ $\PSL_2$-index:$1008$
Genus: $70 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$
Cusps: $30$ (none of which are rational) Cusp widths $28^{24}\cdot56^{6}$ Cusp orbits $3^{4}\cdot6^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $6$
$\Q$-gonality: $10 \le \gamma \le 21$
$\overline{\Q}$-gonality: $10 \le \gamma \le 21$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.2016.70.6438

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}7&12\\12&43\end{bmatrix}$, $\begin{bmatrix}21&16\\54&7\end{bmatrix}$, $\begin{bmatrix}23&20\\6&13\end{bmatrix}$, $\begin{bmatrix}27&28\\0&27\end{bmatrix}$, $\begin{bmatrix}37&16\\4&33\end{bmatrix}$, $\begin{bmatrix}47&16\\44&39\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.1008.70.l.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $1536$

Jacobian

Conductor: $2^{194}\cdot7^{140}$
Simple: no
Squarefree: no
Decomposition: $1^{10}\cdot2^{12}\cdot6^{2}\cdot12^{2}$
Newforms: 98.2.a.b$^{4}$, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.e$^{2}$, 392.2.b.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,11,23,37,67,103,149$, and therefore no rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
56.1008.34-28.b.1.11 $56$ $2$ $2$ $34$ $6$ $6^{2}\cdot12^{2}$
56.1008.34-28.b.1.33 $56$ $2$ $2$ $34$ $6$ $6^{2}\cdot12^{2}$
56.1008.34-56.z.2.18 $56$ $2$ $2$ $34$ $1$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.34-56.z.2.21 $56$ $2$ $2$ $34$ $1$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.34-56.z.2.43 $56$ $2$ $2$ $34$ $1$ $1^{6}\cdot2^{6}\cdot6\cdot12$
56.1008.34-56.z.2.48 $56$ $2$ $2$ $34$ $1$ $1^{6}\cdot2^{6}\cdot6\cdot12$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.4032.139-56.h.1.18 $56$ $2$ $2$ $139$ $16$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.l.1.20 $56$ $2$ $2$ $139$ $30$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.bi.1.13 $56$ $2$ $2$ $139$ $12$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.bm.1.17 $56$ $2$ $2$ $139$ $21$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.cj.1.18 $56$ $2$ $2$ $139$ $19$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.cn.1.19 $56$ $2$ $2$ $139$ $18$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.dh.1.11 $56$ $2$ $2$ $139$ $15$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.139-56.dl.1.18 $56$ $2$ $2$ $139$ $21$ $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pd.1.19 $56$ $2$ $2$ $145$ $23$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pd.2.10 $56$ $2$ $2$ $145$ $23$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pd.2.16 $56$ $2$ $2$ $145$ $23$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pe.1.20 $56$ $2$ $2$ $145$ $24$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pe.2.9 $56$ $2$ $2$ $145$ $24$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pe.2.15 $56$ $2$ $2$ $145$ $24$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pf.1.22 $56$ $2$ $2$ $145$ $21$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pf.2.3 $56$ $2$ $2$ $145$ $21$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pf.2.5 $56$ $2$ $2$ $145$ $21$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pg.1.23 $56$ $2$ $2$ $145$ $18$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pg.2.4 $56$ $2$ $2$ $145$ $18$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pg.2.6 $56$ $2$ $2$ $145$ $18$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.ph.1.22 $56$ $2$ $2$ $145$ $25$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.ph.2.1 $56$ $2$ $2$ $145$ $25$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.ph.2.7 $56$ $2$ $2$ $145$ $25$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pi.1.22 $56$ $2$ $2$ $145$ $26$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pi.2.2 $56$ $2$ $2$ $145$ $26$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pi.2.8 $56$ $2$ $2$ $145$ $26$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.4032.145-56.pj.1.17 $56$ $2$ $2$ $145$ $20$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pj.2.12 $56$ $2$ $2$ $145$ $20$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pj.2.15 $56$ $2$ $2$ $145$ $20$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pk.1.18 $56$ $2$ $2$ $145$ $19$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pk.2.10 $56$ $2$ $2$ $145$ $19$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.145-56.pk.2.13 $56$ $2$ $2$ $145$ $19$ $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.4032.151-56.ex.1.18 $56$ $2$ $2$ $151$ $23$ $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.4032.151-56.ey.1.18 $56$ $2$ $2$ $151$ $19$ $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.4032.151-56.ez.1.19 $56$ $2$ $2$ $151$ $24$ $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.4032.151-56.fa.1.18 $56$ $2$ $2$ $151$ $20$ $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$
56.4032.151-56.fb.1.22 $56$ $2$ $2$ $151$ $18$ $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.4032.151-56.fc.1.23 $56$ $2$ $2$ $151$ $25$ $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.4032.151-56.fd.1.22 $56$ $2$ $2$ $151$ $21$ $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$
56.4032.151-56.fe.1.22 $56$ $2$ $2$ $151$ $26$ $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$