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Properties

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

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
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: $1$
$\Q$-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: 40.48.1.50

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&0\\10&39\end{bmatrix}$, $\begin{bmatrix}19&32\\24&25\end{bmatrix}$, $\begin{bmatrix}27&12\\24&1\end{bmatrix}$, $\begin{bmatrix}27&24\\32&7\end{bmatrix}$, $\begin{bmatrix}31&12\\6&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.bf.1.1, 40.96.1-40.bf.1.2, 40.96.1-40.bf.1.3, 40.96.1-40.bf.1.4, 40.96.1-40.bf.1.5, 40.96.1-40.bf.1.6, 40.96.1-40.bf.1.7, 40.96.1-40.bf.1.8, 40.96.1-40.bf.1.9, 40.96.1-40.bf.1.10, 40.96.1-40.bf.1.11, 40.96.1-40.bf.1.12, 40.96.1-40.bf.1.13, 40.96.1-40.bf.1.14, 40.96.1-40.bf.1.15, 40.96.1-40.bf.1.16, 120.96.1-40.bf.1.1, 120.96.1-40.bf.1.2, 120.96.1-40.bf.1.3, 120.96.1-40.bf.1.4, 120.96.1-40.bf.1.5, 120.96.1-40.bf.1.6, 120.96.1-40.bf.1.7, 120.96.1-40.bf.1.8, 120.96.1-40.bf.1.9, 120.96.1-40.bf.1.10, 120.96.1-40.bf.1.11, 120.96.1-40.bf.1.12, 120.96.1-40.bf.1.13, 120.96.1-40.bf.1.14, 120.96.1-40.bf.1.15, 120.96.1-40.bf.1.16, 280.96.1-40.bf.1.1, 280.96.1-40.bf.1.2, 280.96.1-40.bf.1.3, 280.96.1-40.bf.1.4, 280.96.1-40.bf.1.5, 280.96.1-40.bf.1.6, 280.96.1-40.bf.1.7, 280.96.1-40.bf.1.8, 280.96.1-40.bf.1.9, 280.96.1-40.bf.1.10, 280.96.1-40.bf.1.11, 280.96.1-40.bf.1.12, 280.96.1-40.bf.1.13, 280.96.1-40.bf.1.14, 280.96.1-40.bf.1.15, 280.96.1-40.bf.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 275x + 1750 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{5^4}\cdot\frac{120x^{2}y^{14}-13941250x^{2}y^{12}z^{2}+428401875000x^{2}y^{10}z^{4}-6900508681640625x^{2}y^{8}z^{6}+66088960078125000000x^{2}y^{6}z^{8}-384352800001373291015625x^{2}y^{4}z^{10}+1274845919999771118164062500x^{2}y^{2}z^{12}-1864447160000002384185791015625x^{2}z^{14}-7900xy^{14}z+511725000xy^{12}z^{3}-13152999609375xy^{10}z^{5}+187071728613281250xy^{8}z^{7}-1628934399267578125000xy^{6}z^{9}+8712791999981689453125000xy^{4}z^{11}-26665227200002384185791015625xy^{2}z^{13}+35689500399999976158142089843750xz^{15}-y^{16}+354000y^{14}z^{2}-14006062500y^{12}z^{4}+272547234375000y^{10}z^{6}-3072506682617187500y^{8}z^{8}+21384191985351562500000y^{6}z^{10}-89775007999855041503906250y^{4}z^{12}+203145392000038146972656250000y^{2}z^{14}-170450287999999582767486572265625z^{16}}{z^{2}y^{4}(x^{2}y^{8}-354500x^{2}y^{6}z^{2}+9836015625x^{2}y^{4}z^{4}-74056000000000x^{2}y^{2}z^{6}+161564000000000000x^{2}z^{8}-120xy^{8}z+13575625xy^{6}z^{3}-261559843750xy^{4}z^{5}+1613600000000000xy^{2}z^{7}-3092680000000000000xz^{9}+7650y^{8}z^{2}-361150000y^{6}z^{4}+3942397265625y^{4}z^{6}-14274400000000000y^{2}z^{8}+14770400000000000000z^{10})}$

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.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.i.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.c.1 $40$ $2$ $2$ $1$ $1$ 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.s.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.x.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bf.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bh.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bx.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.bz.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.cc.1 $40$ $2$ $2$ $1$ $1$ dimension zero
40.96.1.cd.2 $40$ $2$ $2$ $1$ $1$ dimension zero
40.240.17.bz.1 $40$ $5$ $5$ $17$ $2$ $1^{6}\cdot2^{5}$
40.288.17.ew.2 $40$ $6$ $6$ $17$ $3$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.iz.1 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{6}\cdot4^{2}$
120.96.1.ga.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ge.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.hh.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.hl.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.mj.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.mn.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.np.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.nt.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.sd.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.jl.1 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.gd.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gf.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gt.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gv.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ip.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ir.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.jf.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.jh.2 $280$ $2$ $2$ $1$ $?$ dimension zero