Properties

Label 8.24.1.bd.1
Level $8$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $32$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $4^{2}\cdot8^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8C1
Rouse and Zureick-Brown (RZB) label: X146
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.24.1.16

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&3\\6&3\end{bmatrix}$, $\begin{bmatrix}5&4\\2&3\end{bmatrix}$, $\begin{bmatrix}7&0\\0&3\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^4:C_4$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 8-isogeny field degree: $4$
Cyclic 8-torsion field degree: $16$
Full 8-torsion field degree: $64$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 11x - 14 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(-2:0:1)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{12x^{2}y^{6}+2787x^{2}y^{4}z^{2}+141696x^{2}y^{2}z^{4}+2054593x^{2}z^{6}+98xy^{6}z+14304xy^{4}z^{3}+604801xy^{2}z^{5}+7865854xz^{7}+y^{8}+528y^{6}z^{2}+39940y^{4}z^{4}+995324y^{2}z^{6}+7513337z^{8}}{z^{2}(x^{2}y^{4}+472x^{2}y^{2}z^{2}+19024x^{2}z^{4}+12xy^{4}z+2384xy^{2}z^{3}+72832xz^{5}+84y^{4}z^{2}+6144y^{2}z^{4}+69568z^{6})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}^+(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
8.12.0.u.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.12.1.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
8.48.1.h.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.w.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.bk.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.48.1.br.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hh.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hl.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.hx.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.48.1.ib.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.5.gr.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
24.96.5.dh.1 $24$ $4$ $4$ $5$ $0$ $1^{4}$
40.48.1.gl.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.gp.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.hb.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.hf.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.120.9.cz.1 $40$ $5$ $5$ $9$ $2$ $1^{6}\cdot2$
40.144.9.fd.1 $40$ $6$ $6$ $9$ $0$ $1^{6}\cdot2$
40.240.17.pr.1 $40$ $10$ $10$ $17$ $5$ $1^{12}\cdot2^{2}$
56.48.1.gj.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.gn.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.gz.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.48.1.hd.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.192.13.dh.1 $56$ $8$ $8$ $13$ $3$ $1^{12}$
56.504.37.gr.1 $56$ $21$ $21$ $37$ $13$ $1^{8}\cdot2^{12}\cdot4$
56.672.49.gr.1 $56$ $28$ $28$ $49$ $16$ $1^{20}\cdot2^{12}\cdot4$
88.48.1.gj.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.gn.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.gz.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.48.1.hd.1 $88$ $2$ $2$ $1$ $?$ dimension zero
88.288.21.dh.1 $88$ $12$ $12$ $21$ $?$ not computed
104.48.1.gl.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.gp.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.hb.1 $104$ $2$ $2$ $1$ $?$ dimension zero
104.48.1.hf.1 $104$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.xj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.xn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.yp.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.yt.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gl.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.gp.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.hb.1 $136$ $2$ $2$ $1$ $?$ dimension zero
136.48.1.hf.1 $136$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gj.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gn.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.gz.1 $152$ $2$ $2$ $1$ $?$ dimension zero
152.48.1.hd.1 $152$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.xh.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.xl.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.yn.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.48.1.yr.1 $168$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gj.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gn.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.gz.1 $184$ $2$ $2$ $1$ $?$ dimension zero
184.48.1.hd.1 $184$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.gl.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.gp.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.hb.1 $232$ $2$ $2$ $1$ $?$ dimension zero
232.48.1.hf.1 $232$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gj.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gn.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.gz.1 $248$ $2$ $2$ $1$ $?$ dimension zero
248.48.1.hd.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.xh.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.xl.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.yn.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.48.1.yr.1 $264$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.wl.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.wp.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.xr.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.48.1.xv.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gl.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.gp.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.hb.1 $296$ $2$ $2$ $1$ $?$ dimension zero
296.48.1.hf.1 $296$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.xj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.xn.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.yp.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.48.1.yt.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gl.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.gp.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.hb.1 $328$ $2$ $2$ $1$ $?$ dimension zero
328.48.1.hf.1 $328$ $2$ $2$ $1$ $?$ dimension zero