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Properties

Label 8.48.1.m.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
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: 8G1
Rouse and Zureick-Brown (RZB) label: X269
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.32

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}1&0\\4&5\end{bmatrix}$, $\begin{bmatrix}3&2\\4&1\end{bmatrix}$, $\begin{bmatrix}5&6\\0&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^2\times D_4$
Contains $-I$: yes
Quadratic refinements: 8.96.1-8.m.1.1, 8.96.1-8.m.1.2, 8.96.1-8.m.1.3, 8.96.1-8.m.1.4, 8.96.1-8.m.1.5, 8.96.1-8.m.1.6, 8.96.1-8.m.1.7, 8.96.1-8.m.1.8, 24.96.1-8.m.1.1, 24.96.1-8.m.1.2, 24.96.1-8.m.1.3, 24.96.1-8.m.1.4, 24.96.1-8.m.1.5, 24.96.1-8.m.1.6, 24.96.1-8.m.1.7, 24.96.1-8.m.1.8, 40.96.1-8.m.1.1, 40.96.1-8.m.1.2, 40.96.1-8.m.1.3, 40.96.1-8.m.1.4, 40.96.1-8.m.1.5, 40.96.1-8.m.1.6, 40.96.1-8.m.1.7, 40.96.1-8.m.1.8, 56.96.1-8.m.1.1, 56.96.1-8.m.1.2, 56.96.1-8.m.1.3, 56.96.1-8.m.1.4, 56.96.1-8.m.1.5, 56.96.1-8.m.1.6, 56.96.1-8.m.1.7, 56.96.1-8.m.1.8, 88.96.1-8.m.1.1, 88.96.1-8.m.1.2, 88.96.1-8.m.1.3, 88.96.1-8.m.1.4, 88.96.1-8.m.1.5, 88.96.1-8.m.1.6, 88.96.1-8.m.1.7, 88.96.1-8.m.1.8, 104.96.1-8.m.1.1, 104.96.1-8.m.1.2, 104.96.1-8.m.1.3, 104.96.1-8.m.1.4, 104.96.1-8.m.1.5, 104.96.1-8.m.1.6, 104.96.1-8.m.1.7, 104.96.1-8.m.1.8, 120.96.1-8.m.1.1, 120.96.1-8.m.1.2, 120.96.1-8.m.1.3, 120.96.1-8.m.1.4, 120.96.1-8.m.1.5, 120.96.1-8.m.1.6, 120.96.1-8.m.1.7, 120.96.1-8.m.1.8, 136.96.1-8.m.1.1, 136.96.1-8.m.1.2, 136.96.1-8.m.1.3, 136.96.1-8.m.1.4, 136.96.1-8.m.1.5, 136.96.1-8.m.1.6, 136.96.1-8.m.1.7, 136.96.1-8.m.1.8, 152.96.1-8.m.1.1, 152.96.1-8.m.1.2, 152.96.1-8.m.1.3, 152.96.1-8.m.1.4, 152.96.1-8.m.1.5, 152.96.1-8.m.1.6, 152.96.1-8.m.1.7, 152.96.1-8.m.1.8, 168.96.1-8.m.1.1, 168.96.1-8.m.1.2, 168.96.1-8.m.1.3, 168.96.1-8.m.1.4, 168.96.1-8.m.1.5, 168.96.1-8.m.1.6, 168.96.1-8.m.1.7, 168.96.1-8.m.1.8, 184.96.1-8.m.1.1, 184.96.1-8.m.1.2, 184.96.1-8.m.1.3, 184.96.1-8.m.1.4, 184.96.1-8.m.1.5, 184.96.1-8.m.1.6, 184.96.1-8.m.1.7, 184.96.1-8.m.1.8, 232.96.1-8.m.1.1, 232.96.1-8.m.1.2, 232.96.1-8.m.1.3, 232.96.1-8.m.1.4, 232.96.1-8.m.1.5, 232.96.1-8.m.1.6, 232.96.1-8.m.1.7, 232.96.1-8.m.1.8, 248.96.1-8.m.1.1, 248.96.1-8.m.1.2, 248.96.1-8.m.1.3, 248.96.1-8.m.1.4, 248.96.1-8.m.1.5, 248.96.1-8.m.1.6, 248.96.1-8.m.1.7, 248.96.1-8.m.1.8, 264.96.1-8.m.1.1, 264.96.1-8.m.1.2, 264.96.1-8.m.1.3, 264.96.1-8.m.1.4, 264.96.1-8.m.1.5, 264.96.1-8.m.1.6, 264.96.1-8.m.1.7, 264.96.1-8.m.1.8, 280.96.1-8.m.1.1, 280.96.1-8.m.1.2, 280.96.1-8.m.1.3, 280.96.1-8.m.1.4, 280.96.1-8.m.1.5, 280.96.1-8.m.1.6, 280.96.1-8.m.1.7, 280.96.1-8.m.1.8, 296.96.1-8.m.1.1, 296.96.1-8.m.1.2, 296.96.1-8.m.1.3, 296.96.1-8.m.1.4, 296.96.1-8.m.1.5, 296.96.1-8.m.1.6, 296.96.1-8.m.1.7, 296.96.1-8.m.1.8, 312.96.1-8.m.1.1, 312.96.1-8.m.1.2, 312.96.1-8.m.1.3, 312.96.1-8.m.1.4, 312.96.1-8.m.1.5, 312.96.1-8.m.1.6, 312.96.1-8.m.1.7, 312.96.1-8.m.1.8, 328.96.1-8.m.1.1, 328.96.1-8.m.1.2, 328.96.1-8.m.1.3, 328.96.1-8.m.1.4, 328.96.1-8.m.1.5, 328.96.1-8.m.1.6, 328.96.1-8.m.1.7, 328.96.1-8.m.1.8
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $4$
Full 8-torsion field degree: $32$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 44x - 112 $
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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
$(-4:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4}\cdot\frac{48x^{2}y^{14}+356896x^{2}y^{12}z^{2}+701893632x^{2}y^{10}z^{4}+723570779136x^{2}y^{8}z^{6}+443515503378432x^{2}y^{6}z^{8}+165078270613192704x^{2}y^{4}z^{10}+35042697816563515392x^{2}y^{2}z^{12}+3279970130870308700160x^{2}z^{14}+1264xy^{14}z+5240064xy^{12}z^{3}+8619949824xy^{10}z^{5}+7846356996096xy^{8}z^{7}+4372637482614784xy^{6}z^{9}+1496846267870871552xy^{4}z^{11}+293187273636914921472xy^{2}z^{13}+25114253234762353213440xz^{15}+y^{16}+22656y^{14}z^{2}+57368832y^{12}z^{4}+71446622208y^{10}z^{6}+51548108275712y^{8}z^{8}+22961101307117568y^{6}z^{10}+6169291573720252416y^{4}z^{12}+893442882532622204928y^{2}z^{14}+47977490845124490428416z^{16}}{z^{2}y^{4}(x^{2}y^{8}+22688x^{2}y^{6}z^{2}+40288320x^{2}y^{4}z^{4}+19413336064x^{2}y^{2}z^{6}+2710594125824x^{2}z^{8}+48xy^{8}z+347536xy^{6}z^{3}+428539648xy^{4}z^{5}+169198223360xy^{2}z^{7}+20754624151552xz^{9}+1224y^{8}z^{2}+3698176y^{6}z^{4}+2583689472y^{4}z^{6}+598711730176y^{2}z^{8}+39648990593024z^{10})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.d.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.e.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.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
8.96.1.a.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.f.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.i.2 $8$ $2$ $2$ $1$ $0$ dimension zero
8.96.1.j.2 $8$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bc.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.be.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bk.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bm.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.ee.1 $24$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
24.192.9.ck.1 $24$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
40.96.1.bc.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.be.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bk.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bm.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bm.1 $40$ $5$ $5$ $17$ $4$ $1^{6}\cdot2^{5}$
40.288.17.cu.2 $40$ $6$ $6$ $17$ $0$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.gg.1 $40$ $10$ $10$ $33$ $6$ $1^{12}\cdot2^{6}\cdot4^{2}$
56.96.1.bc.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.be.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bk.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bm.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.ck.2 $56$ $8$ $8$ $25$ $3$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.ee.1 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.ee.2 $56$ $28$ $28$ $97$ $12$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
88.96.1.bc.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.be.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.bk.2 $88$ $2$ $2$ $1$ $?$ dimension zero
88.96.1.bm.2 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.bc.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.be.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.bk.2 $104$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.bm.2 $104$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dy.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ea.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.eo.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.eq.2 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.bc.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.be.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.bk.2 $136$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.bm.2 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.bc.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.be.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.bk.2 $152$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.bm.2 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.dy.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ea.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.eo.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.eq.2 $168$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.bc.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.be.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.bk.2 $184$ $2$ $2$ $1$ $?$ dimension zero
184.96.1.bm.2 $184$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.bc.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.be.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.bk.2 $232$ $2$ $2$ $1$ $?$ dimension zero
232.96.1.bm.2 $232$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.bc.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.be.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.bk.2 $248$ $2$ $2$ $1$ $?$ dimension zero
248.96.1.bm.2 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.dy.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ea.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.eo.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.eq.2 $264$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dy.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ea.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.eo.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.eq.2 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.bc.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.be.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.bk.2 $296$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.bm.2 $296$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.dy.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ea.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.eo.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.eq.2 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.bc.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.be.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.bk.2 $328$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.bm.2 $328$ $2$ $2$ $1$ $?$ dimension zero