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Properties

Label 40.48.1.bi.2
Level $40$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $1600$
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, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.81

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}15&36\\34&15\end{bmatrix}$, $\begin{bmatrix}17&28\\18&15\end{bmatrix}$, $\begin{bmatrix}25&12\\18&33\end{bmatrix}$, $\begin{bmatrix}31&24\\30&23\end{bmatrix}$, $\begin{bmatrix}33&36\\22&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.bi.2.1, 40.96.1-40.bi.2.2, 40.96.1-40.bi.2.3, 40.96.1-40.bi.2.4, 40.96.1-40.bi.2.5, 40.96.1-40.bi.2.6, 40.96.1-40.bi.2.7, 40.96.1-40.bi.2.8, 40.96.1-40.bi.2.9, 40.96.1-40.bi.2.10, 40.96.1-40.bi.2.11, 40.96.1-40.bi.2.12, 40.96.1-40.bi.2.13, 40.96.1-40.bi.2.14, 40.96.1-40.bi.2.15, 40.96.1-40.bi.2.16, 120.96.1-40.bi.2.1, 120.96.1-40.bi.2.2, 120.96.1-40.bi.2.3, 120.96.1-40.bi.2.4, 120.96.1-40.bi.2.5, 120.96.1-40.bi.2.6, 120.96.1-40.bi.2.7, 120.96.1-40.bi.2.8, 120.96.1-40.bi.2.9, 120.96.1-40.bi.2.10, 120.96.1-40.bi.2.11, 120.96.1-40.bi.2.12, 120.96.1-40.bi.2.13, 120.96.1-40.bi.2.14, 120.96.1-40.bi.2.15, 120.96.1-40.bi.2.16, 280.96.1-40.bi.2.1, 280.96.1-40.bi.2.2, 280.96.1-40.bi.2.3, 280.96.1-40.bi.2.4, 280.96.1-40.bi.2.5, 280.96.1-40.bi.2.6, 280.96.1-40.bi.2.7, 280.96.1-40.bi.2.8, 280.96.1-40.bi.2.9, 280.96.1-40.bi.2.10, 280.96.1-40.bi.2.11, 280.96.1-40.bi.2.12, 280.96.1-40.bi.2.13, 280.96.1-40.bi.2.14, 280.96.1-40.bi.2.15, 280.96.1-40.bi.2.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.n

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 1100x - 14000 $
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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
$(0:1:0)$, $(-20:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^2\cdot5^2}\cdot\frac{240x^{2}y^{14}+4247860000x^{2}y^{12}z^{2}+10115366640000000x^{2}y^{10}z^{4}+5885785344090000000000x^{2}y^{8}z^{6}+1099369357013760000000000000x^{2}y^{6}z^{8}+83657641426951290000000000000000x^{2}y^{4}z^{10}+2737710766919516040000000000000000000x^{2}y^{2}z^{12}+32030958309280317450000000000000000000000x^{2}z^{14}+103600xy^{14}z+658471200000xy^{12}z^{3}+1015750239900000000xy^{10}z^{5}+409934447956200000000000xy^{8}z^{7}+60586848786325600000000000000xy^{6}z^{9}+3930478117650285600000000000000000xy^{4}z^{11}+114526278764409653700000000000000000000xy^{2}z^{13}+1226281896228631347000000000000000000000000xz^{15}+y^{16}+20112000y^{14}z^{2}+86989668000000y^{12}z^{4}+78616007544000000000y^{10}z^{6}+20141247406412000000000000y^{8}z^{8}+2007992773878816000000000000000y^{6}z^{10}+89258362532785194000000000000000000y^{4}z^{12}+1745005629946200144000000000000000000000y^{2}z^{14}+11713254600860499961000000000000000000000000z^{16}}{zy^{4}(28700x^{2}y^{8}z+6680000000x^{2}y^{6}z^{3}+229382100000000x^{2}y^{4}z^{5}+400000000000x^{2}y^{2}z^{7}+100000000000000x^{2}z^{9}+xy^{10}+2390000xy^{8}z^{2}+325308000000xy^{6}z^{4}+8781712000000000xy^{4}z^{6}-7000000000000xy^{2}z^{8}-2000000000000000xz^{10}+240y^{10}z+130460000y^{8}z^{3}+7769760000000y^{6}z^{5}+83881420000000000y^{4}z^{7}-320000000000000y^{2}z^{9}-70000000000000000z^{11})}$

Modular covers

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Cover information

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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.e.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.h.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.d.1 $40$ $2$ $2$ $1$ $0$ 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
40.96.1.g.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.w.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bl.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bp.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bu.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.by.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.cf.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.ch.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.cc.2 $40$ $5$ $5$ $17$ $2$ $1^{6}\cdot2^{5}$
40.288.17.fd.2 $40$ $6$ $6$ $17$ $2$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.jf.2 $40$ $10$ $10$ $33$ $2$ $1^{12}\cdot2^{6}\cdot4^{2}$
120.96.1.gm.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gs.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ht.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.hz.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.mu.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.na.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.oa.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.og.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.sp.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.jr.2 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.hc.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.hg.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.hs.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.hw.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.jo.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.js.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ke.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ki.1 $280$ $2$ $2$ $1$ $?$ dimension zero