Properties

Label 56.2016.145.ph.2
Level $56$
Index $2016$
Genus $145$
Analytic rank $25$
Cusps $48$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $56$ $\SL_2$-level: $56$ Newform level: $3136$
Index: $2016$ $\PSL_2$-index:$2016$
Genus: $145 = 1 + \frac{ 2016 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$
Cusps: $48$ (none of which are rational) Cusp widths $28^{24}\cdot56^{24}$ Cusp orbits $6^{8}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $25$
$\Q$-gonality: $20 \le \gamma \le 42$
$\overline{\Q}$-gonality: $20 \le \gamma \le 42$
Rational cusps: $0$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.2016.145.4662

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}19&4\\46&37\end{bmatrix}$, $\begin{bmatrix}23&8\\50&33\end{bmatrix}$, $\begin{bmatrix}27&28\\0&55\end{bmatrix}$, $\begin{bmatrix}39&8\\18&45\end{bmatrix}$, $\begin{bmatrix}47&8\\0&23\end{bmatrix}$, $\begin{bmatrix}49&8\\6&35\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.4032.145-56.ph.2.1, 56.4032.145-56.ph.2.2, 56.4032.145-56.ph.2.3, 56.4032.145-56.ph.2.4, 56.4032.145-56.ph.2.5, 56.4032.145-56.ph.2.6, 56.4032.145-56.ph.2.7, 56.4032.145-56.ph.2.8, 56.4032.145-56.ph.2.9, 56.4032.145-56.ph.2.10, 56.4032.145-56.ph.2.11, 56.4032.145-56.ph.2.12, 56.4032.145-56.ph.2.13, 56.4032.145-56.ph.2.14, 56.4032.145-56.ph.2.15, 56.4032.145-56.ph.2.16, 56.4032.145-56.ph.2.17, 56.4032.145-56.ph.2.18, 56.4032.145-56.ph.2.19, 56.4032.145-56.ph.2.20, 56.4032.145-56.ph.2.21, 56.4032.145-56.ph.2.22, 56.4032.145-56.ph.2.23, 56.4032.145-56.ph.2.24
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $1536$

Jacobian

Conductor: $2^{572}\cdot7^{264}$
Simple: no
Squarefree: no
Decomposition: $1^{33}\cdot2^{24}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
Newforms: 56.2.b.a, 56.2.b.b, 64.2.a.a, 98.2.a.b$^{4}$, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a, 224.2.b.b, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.a, 392.2.b.d, 392.2.b.e$^{3}$, 392.2.b.g$^{2}$, 448.2.a.a, 448.2.a.b, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.f, 448.2.a.g, 448.2.a.h, 448.2.a.i, 448.2.a.j, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.b, 1568.2.b.c, 1568.2.b.f, 3136.2.a.b, 3136.2.a.ba, 3136.2.a.bc, 3136.2.a.bd, 3136.2.a.bh, 3136.2.a.bq, 3136.2.a.bt, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.d, 3136.2.a.g, 3136.2.a.h, 3136.2.a.j, 3136.2.a.l, 3136.2.a.n, 3136.2.a.o, 3136.2.a.r, 3136.2.a.s, 3136.2.a.u, 3136.2.a.x

Rational points

This modular curve has no $\Q_p$ points for $p=3,5,11,23,37,67,103,149,233$, 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
56.1008.70.f.1 $56$ $2$ $2$ $70$ $6$ $1^{23}\cdot2^{8}\cdot6^{2}\cdot12^{2}$
56.1008.70.l.1 $56$ $2$ $2$ $70$ $6$ $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$
56.1008.70.cr.2 $56$ $2$ $2$ $70$ $9$ $1^{15}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.70.ct.2 $56$ $2$ $2$ $70$ $9$ $1^{15}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.73.cw.1 $56$ $2$ $2$ $73$ $25$ $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{2}$
56.1008.73.jt.2 $56$ $2$ $2$ $73$ $12$ $1^{20}\cdot2^{10}\cdot4^{2}\cdot6^{2}\cdot12$
56.1008.73.jv.2 $56$ $2$ $2$ $73$ $12$ $1^{20}\cdot2^{10}\cdot4^{2}\cdot6^{2}\cdot12$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
56.4032.289.gs.2 $56$ $2$ $2$ $289$ $53$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.hn.2 $56$ $2$ $2$ $289$ $63$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.mt.1 $56$ $2$ $2$ $289$ $48$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.nu.1 $56$ $2$ $2$ $289$ $53$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.wg.1 $56$ $2$ $2$ $289$ $51$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.wx.1 $56$ $2$ $2$ $289$ $54$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.bak.2 $56$ $2$ $2$ $289$ $48$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.289.bbe.1 $56$ $2$ $2$ $289$ $58$ $1^{34}\cdot2^{21}\cdot4^{5}\cdot6^{4}\cdot12^{2}$
56.4032.301.rg.1 $56$ $2$ $2$ $301$ $61$ $1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rl.2 $56$ $2$ $2$ $301$ $50$ $1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rp.2 $56$ $2$ $2$ $301$ $63$ $1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rr.2 $56$ $2$ $2$ $301$ $54$ $1^{30}\cdot2^{27}\cdot4^{8}\cdot6^{2}\cdot12\cdot16$
56.4032.301.rw.2 $56$ $2$ $2$ $301$ $61$ $1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.sa.4 $56$ $2$ $2$ $301$ $50$ $1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.se.2 $56$ $2$ $2$ $301$ $63$ $1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$
56.4032.301.sf.2 $56$ $2$ $2$ $301$ $54$ $1^{30}\cdot2^{25}\cdot4^{5}\cdot6^{2}\cdot8\cdot12^{3}$