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Properties

Label 20.72.1.k.1
Level $20$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $400$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4$
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: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.72.1.17

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}5&3\\12&5\end{bmatrix}$, $\begin{bmatrix}11&13\\4&5\end{bmatrix}$, $\begin{bmatrix}17&8\\6&5\end{bmatrix}$, $\begin{bmatrix}19&2\\6&15\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $D_{10}.(C_4\times D_4)$
Contains $-I$: yes
Quadratic refinements: 20.144.1-20.k.1.1, 20.144.1-20.k.1.2, 20.144.1-20.k.1.3, 20.144.1-20.k.1.4, 20.144.1-20.k.1.5, 20.144.1-20.k.1.6, 20.144.1-20.k.1.7, 20.144.1-20.k.1.8, 40.144.1-20.k.1.1, 40.144.1-20.k.1.2, 40.144.1-20.k.1.3, 40.144.1-20.k.1.4, 40.144.1-20.k.1.5, 40.144.1-20.k.1.6, 40.144.1-20.k.1.7, 40.144.1-20.k.1.8, 60.144.1-20.k.1.1, 60.144.1-20.k.1.2, 60.144.1-20.k.1.3, 60.144.1-20.k.1.4, 60.144.1-20.k.1.5, 60.144.1-20.k.1.6, 60.144.1-20.k.1.7, 60.144.1-20.k.1.8, 120.144.1-20.k.1.1, 120.144.1-20.k.1.2, 120.144.1-20.k.1.3, 120.144.1-20.k.1.4, 120.144.1-20.k.1.5, 120.144.1-20.k.1.6, 120.144.1-20.k.1.7, 120.144.1-20.k.1.8, 140.144.1-20.k.1.1, 140.144.1-20.k.1.2, 140.144.1-20.k.1.3, 140.144.1-20.k.1.4, 140.144.1-20.k.1.5, 140.144.1-20.k.1.6, 140.144.1-20.k.1.7, 140.144.1-20.k.1.8, 220.144.1-20.k.1.1, 220.144.1-20.k.1.2, 220.144.1-20.k.1.3, 220.144.1-20.k.1.4, 220.144.1-20.k.1.5, 220.144.1-20.k.1.6, 220.144.1-20.k.1.7, 220.144.1-20.k.1.8, 260.144.1-20.k.1.1, 260.144.1-20.k.1.2, 260.144.1-20.k.1.3, 260.144.1-20.k.1.4, 260.144.1-20.k.1.5, 260.144.1-20.k.1.6, 260.144.1-20.k.1.7, 260.144.1-20.k.1.8, 280.144.1-20.k.1.1, 280.144.1-20.k.1.2, 280.144.1-20.k.1.3, 280.144.1-20.k.1.4, 280.144.1-20.k.1.5, 280.144.1-20.k.1.6, 280.144.1-20.k.1.7, 280.144.1-20.k.1.8
Cyclic 20-isogeny field degree: $2$
Cyclic 20-torsion field degree: $16$
Full 20-torsion field degree: $640$

Jacobian

Conductor: $2^{4}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 400.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 33x - 62 $
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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
$(-2:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^2}\cdot\frac{120x^{2}y^{22}+206485000x^{2}y^{20}z^{2}+32210190625000x^{2}y^{18}z^{4}+597476187421875000x^{2}y^{16}z^{6}+2283090346386718750000x^{2}y^{14}z^{8}+3130825629816894531250000x^{2}y^{12}z^{10}+2035533957219543457031250000x^{2}y^{10}z^{12}+719427479791831970214843750000x^{2}y^{8}z^{14}+146568345368573665618896484375000x^{2}y^{6}z^{16}+17217416825558245182037353515625000x^{2}y^{4}z^{18}+1084288214769847691059112548828125000x^{2}y^{2}z^{20}+28368418725221417844295501708984375000x^{2}z^{22}+24480xy^{22}z+13205590000xy^{20}z^{3}+1099673356250000xy^{18}z^{5}+11476004773125000000xy^{16}z^{7}+31476256450000000000000xy^{14}z^{9}+35166792612895507812500000xy^{12}z^{11}+19919212983715209960937500000xy^{10}z^{13}+6374238292458343505859375000000xy^{8}z^{15}+1204365980815687179565429687500000xy^{6}z^{17}+133323283080276846885681152343750000xy^{4}z^{19}+8001421747394576668739318847656250000xy^{2}z^{21}+201136884925290942192077636718750000000xz^{23}+y^{24}+2129980y^{22}z^{2}+741199021250y^{20}z^{4}+28006755442187500y^{18}z^{6}+165149691870537109375y^{16}z^{8}+301040044507373046875000y^{14}z^{10}+242922496438685913085937500y^{12}z^{12}+103859588178999481201171875000y^{10}z^{14}+25674510415187585353851318359375y^{8}z^{16}+3785330045593746900558471679687500y^{6}z^{18}+327010411639093235135078430175781250y^{4}z^{20}+15172219036687370389699935913085937500y^{2}z^{22}+288800094949710764922201633453369140625z^{24}}{zy^{4}(4400x^{2}y^{16}z-2175000x^{2}y^{14}z^{3}+616796875x^{2}y^{12}z^{5}+108300781250x^{2}y^{10}z^{7}+5279541015625x^{2}y^{8}z^{9}+21362304687500x^{2}y^{6}z^{11}-5626678466796875x^{2}y^{4}z^{13}-166893005371093750x^{2}y^{2}z^{15}-1490116119384765625x^{2}z^{17}+xy^{18}+87725xy^{16}z^{2}-23137500xy^{14}z^{4}-4310156250xy^{12}z^{6}-239648437500xy^{10}z^{8}-2746582031250xy^{8}z^{10}+268554687500000xy^{6}z^{12}+12779235839843750xy^{4}z^{14}+226497650146484375xy^{2}z^{16}+1490116119384765625xz^{18}+107y^{18}z+567225y^{16}z^{3}+197737500y^{14}z^{5}-7659765625y^{12}z^{7}-3224511718750y^{10}z^{9}-185607910156250y^{8}z^{11}-1837158203125000y^{6}z^{13}+150585174560546875y^{4}z^{15}+4994869232177734375y^{2}z^{17}+46193599700927734375z^{19})}$

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
10.36.0.a.1 $10$ $2$ $2$ $0$ $0$ full Jacobian
20.36.0.c.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
20.36.1.e.1 $20$ $2$ $2$ $1$ $0$ 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
20.144.5.e.2 $20$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
20.144.5.k.2 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.ba.2 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.bc.2 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.360.13.p.1 $20$ $5$ $5$ $13$ $1$ $1^{6}\cdot2^{3}$
40.144.5.bg.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.cv.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ii.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.iy.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.hk.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.ho.2 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.pa.2 $60$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
60.144.5.pe.2 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.216.13.fw.1 $60$ $3$ $3$ $13$ $0$ $1^{6}\cdot2^{3}$
60.288.13.ni.1 $60$ $4$ $4$ $13$ $1$ $1^{6}\cdot2^{3}$
100.360.13.k.1 $100$ $5$ $5$ $13$ $?$ not computed
120.144.5.chf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cih.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehs.2 $120$ $2$ $2$ $5$ $?$ not computed
140.144.5.ea.2 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.ec.2 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.fg.2 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.fi.2 $140$ $2$ $2$ $5$ $?$ not computed
220.144.5.ea.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.ec.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.fg.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.fi.2 $220$ $2$ $2$ $5$ $?$ not computed
260.144.5.ea.2 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.ec.2 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.fg.2 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.fi.2 $260$ $2$ $2$ $5$ $?$ not computed
280.144.5.bft.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgh.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.boi.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bow.2 $280$ $2$ $2$ $5$ $?$ not computed