Properties

Label 49.115248.4215-49.k.1.2
Level $49$
Index $115248$
Genus $4215$
Analytic rank $105$
Cusps $1176$
$\Q$-cusps $21$

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Invariants

Level: $49$ $\SL_2$-level: $49$ Newform level: $2401$
Index: $115248$ $\PSL_2$-index:$57624$
Genus: $4215 = 1 + \frac{ 57624 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 1176 }{2}$
Cusps: $1176$ (of which $21$ are rational) Cusp widths $49^{1176}$ Cusp orbits $1^{21}\cdot6^{21}\cdot21\cdot42^{24}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $105$
$\Q$-gonality: $572 \le \gamma \le 1029$
$\overline{\Q}$-gonality: $572 \le \gamma \le 1029$
Rational cusps: $21$
Rational CM points: none

Other labels

Rouse, Sutherland, and Zureick-Brown (RSZB) label: 49.115248.4215.1

Level structure

$\GL_2(\Z/49\Z)$-generators: $\begin{bmatrix}47&0\\0&1\end{bmatrix}$
$\GL_2(\Z/49\Z)$-subgroup: $C_{42}$
Contains $-I$: no $\quad$ (see 49.57624.4215.k.1 for the level structure with $-I$)
Cyclic 49-isogeny field degree: $1$
Cyclic 49-torsion field degree: $1$
Full 49-torsion field degree: $42$

Jacobian

Conductor: $7^{15642}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2^{3}\cdot3^{6}\cdot6^{30}\cdot9^{2}\cdot12^{57}\cdot18^{9}\cdot24^{11}\cdot36^{10}\cdot48^{28}\cdot72^{3}\cdot96^{4}\cdot192^{3}$
Newforms: 49.2.a.a$^{3}$, 49.2.c.a$^{3}$, 49.2.e.a$^{3}$, 49.2.e.b$^{3}$, 49.2.g.a$^{3}$, 343.2.a.a$^{2}$, 343.2.a.b$^{2}$, 343.2.a.c$^{2}$, 343.2.a.d$^{2}$, 343.2.a.e$^{2}$, 343.2.c.a$^{2}$, 343.2.c.b$^{2}$, 343.2.c.c$^{2}$, 343.2.c.d$^{2}$, 343.2.c.e$^{2}$, 343.2.e.a$^{2}$, 343.2.e.b$^{2}$, 343.2.e.c$^{2}$, 343.2.e.d$^{2}$, 343.2.g.a$^{2}$, 343.2.g.b$^{2}$, 343.2.g.c$^{2}$, 343.2.g.d$^{2}$, 343.2.g.e$^{2}$, 343.2.g.f$^{2}$, 343.2.g.g$^{2}$, 343.2.g.h$^{2}$, 343.2.g.i$^{2}$, 2401.2.a.a, 2401.2.a.b, 2401.2.a.c, 2401.2.a.d, 2401.2.a.e, 2401.2.a.f, 2401.2.a.g, 2401.2.a.h, 2401.2.a.i, 2401.2.a.j, 2401.2.c.a, 2401.2.c.b, 2401.2.c.c, 2401.2.c.d, 2401.2.c.e, 2401.2.c.f, 2401.2.c.g, 2401.2.c.h, 2401.2.c.i, 2401.2.c.j, 2401.2.e.a, 2401.2.e.b, 2401.2.e.ba, 2401.2.e.bb, 2401.2.e.bc, 2401.2.e.bd, 2401.2.e.be, 2401.2.e.bf, 2401.2.e.bg, 2401.2.e.bh, 2401.2.e.bi, 2401.2.e.bj, 2401.2.e.bk, 2401.2.e.bl, 2401.2.e.bm, 2401.2.e.bn, 2401.2.e.c, 2401.2.e.d, 2401.2.e.e, 2401.2.e.f, 2401.2.e.g, 2401.2.e.h, 2401.2.e.i, 2401.2.e.j, 2401.2.e.k, 2401.2.e.l, 2401.2.e.m, 2401.2.e.n, 2401.2.e.o, 2401.2.e.p, 2401.2.e.q, 2401.2.e.r, 2401.2.e.s, 2401.2.e.t, 2401.2.e.u, 2401.2.e.v, 2401.2.e.w, 2401.2.e.x, 2401.2.e.y, 2401.2.e.z, 2401.2.g.a, 2401.2.g.b, 2401.2.g.ba, 2401.2.g.bb, 2401.2.g.bc, 2401.2.g.bd, 2401.2.g.be, 2401.2.g.bf, 2401.2.g.bg, 2401.2.g.bh, 2401.2.g.bi, 2401.2.g.bj, 2401.2.g.bk, 2401.2.g.bl, 2401.2.g.bm, 2401.2.g.bn, 2401.2.g.bo, 2401.2.g.bp, 2401.2.g.bq, 2401.2.g.br, 2401.2.g.bs, 2401.2.g.bt, 2401.2.g.bu, 2401.2.g.bv, 2401.2.g.c, 2401.2.g.d, 2401.2.g.e, 2401.2.g.f, 2401.2.g.g, 2401.2.g.h, 2401.2.g.i, 2401.2.g.j, 2401.2.g.k, 2401.2.g.l, 2401.2.g.m, 2401.2.g.n, 2401.2.g.o, 2401.2.g.p, 2401.2.g.q, 2401.2.g.r, 2401.2.g.s, 2401.2.g.t, 2401.2.g.u, 2401.2.g.v, 2401.2.g.w, 2401.2.g.x, 2401.2.g.y, 2401.2.g.z

Rational points

This modular curve has 21 rational cusps but no known non-cuspidal rational points.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
49.16464.540-49.k.1.1 $49$ $7$ $7$ $540$ $9$ $1\cdot2\cdot3^{4}\cdot6^{22}\cdot9^{2}\cdot12^{45}\cdot18^{9}\cdot24^{11}\cdot36^{10}\cdot48^{21}\cdot72^{3}\cdot96^{4}\cdot192^{3}$
49.16464.540-49.k.2.1 $49$ $7$ $7$ $540$ $9$ $1\cdot2\cdot3^{4}\cdot6^{22}\cdot9^{2}\cdot12^{45}\cdot18^{9}\cdot24^{11}\cdot36^{10}\cdot48^{21}\cdot72^{3}\cdot96^{4}\cdot192^{3}$
49.16464.603-49.j.1.1 $49$ $7$ $7$ $603$ $87$ $6^{16}\cdot12^{49}\cdot18^{6}\cdot24^{9}\cdot36^{9}\cdot48^{25}\cdot72^{3}\cdot96^{3}\cdot192^{3}$
49.38416.1401-49.e.1.1 $49$ $3$ $3$ $1401$ $93$ $2^{3}\cdot6^{6}\cdot12^{39}\cdot18^{2}\cdot24^{9}\cdot36^{7}\cdot48^{19}\cdot72^{3}\cdot96\cdot192^{3}$