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Properties

Label 68.216.15.l.1
Level $68$
Index $216$
Genus $15$
Analytic rank $2$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $68$ $\SL_2$-level: $68$ Newform level: $272$
Index: $216$ $\PSL_2$-index:$216$
Genus: $15 = 1 + \frac{ 216 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot34^{2}\cdot68^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $6 \le \gamma \le 8$
$\overline{\Q}$-gonality: $6 \le \gamma \le 8$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 68D15
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 68.216.15.4

Level structure

$\GL_2(\Z/68\Z)$-generators: $\begin{bmatrix}13&31\\0&7\end{bmatrix}$, $\begin{bmatrix}15&67\\0&37\end{bmatrix}$, $\begin{bmatrix}25&38\\0&45\end{bmatrix}$, $\begin{bmatrix}31&31\\0&65\end{bmatrix}$, $\begin{bmatrix}63&57\\0&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 68.432.15-68.l.1.1, 68.432.15-68.l.1.2, 68.432.15-68.l.1.3, 68.432.15-68.l.1.4, 68.432.15-68.l.1.5, 68.432.15-68.l.1.6, 68.432.15-68.l.1.7, 68.432.15-68.l.1.8, 136.432.15-68.l.1.1, 136.432.15-68.l.1.2, 136.432.15-68.l.1.3, 136.432.15-68.l.1.4, 136.432.15-68.l.1.5, 136.432.15-68.l.1.6, 136.432.15-68.l.1.7, 136.432.15-68.l.1.8, 136.432.15-68.l.1.9, 136.432.15-68.l.1.10, 136.432.15-68.l.1.11, 136.432.15-68.l.1.12, 136.432.15-68.l.1.13, 136.432.15-68.l.1.14, 136.432.15-68.l.1.15, 136.432.15-68.l.1.16, 136.432.15-68.l.1.17, 136.432.15-68.l.1.18, 136.432.15-68.l.1.19, 136.432.15-68.l.1.20, 136.432.15-68.l.1.21, 136.432.15-68.l.1.22, 136.432.15-68.l.1.23, 136.432.15-68.l.1.24, 204.432.15-68.l.1.1, 204.432.15-68.l.1.2, 204.432.15-68.l.1.3, 204.432.15-68.l.1.4, 204.432.15-68.l.1.5, 204.432.15-68.l.1.6, 204.432.15-68.l.1.7, 204.432.15-68.l.1.8
Cyclic 68-isogeny field degree: $1$
Cyclic 68-torsion field degree: $32$
Full 68-torsion field degree: $34816$

Jacobian

Conductor: $2^{38}\cdot17^{15}$
Simple: no
Squarefree: no
Decomposition: $1^{9}\cdot2^{3}$
Newforms: 17.2.a.a$^{3}$, 34.2.a.a$^{2}$, 68.2.a.a, 272.2.a.a, 272.2.a.b, 272.2.a.c, 272.2.a.d, 272.2.a.e, 272.2.a.f

Rational points

This modular curve has 4 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
4.12.0.d.1 $4$ $18$ $18$ $0$ $0$ full Jacobian
$X_0(17)$ $17$ $12$ $12$ $1$ $0$ $1^{8}\cdot2^{3}$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
4.12.0.d.1 $4$ $18$ $18$ $0$ $0$ full Jacobian
68.108.7.a.1 $68$ $2$ $2$ $7$ $1$ $1^{4}\cdot2^{2}$
$X_0(68)$ $68$ $2$ $2$ $7$ $0$ $1^{4}\cdot2^{2}$
68.108.7.o.1 $68$ $2$ $2$ $7$ $1$ $1^{4}\cdot2^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
68.432.29.t.1 $68$ $2$ $2$ $29$ $2$ $2^{7}$
68.432.29.t.2 $68$ $2$ $2$ $29$ $2$ $2^{7}$
68.432.29.x.1 $68$ $2$ $2$ $29$ $2$ $2^{7}$
68.432.29.x.2 $68$ $2$ $2$ $29$ $2$ $2^{7}$
68.432.31.l.1 $68$ $2$ $2$ $31$ $5$ $1^{8}\cdot2^{4}$
68.432.31.m.1 $68$ $2$ $2$ $31$ $9$ $1^{8}\cdot2^{4}$
68.432.31.n.1 $68$ $2$ $2$ $31$ $2$ $2^{8}$
68.432.31.n.2 $68$ $2$ $2$ $31$ $2$ $2^{8}$
68.3672.271.bi.1 $68$ $17$ $17$ $271$ $107$ $1^{20}\cdot2^{38}\cdot3^{28}\cdot4^{13}\cdot6^{2}\cdot12$