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Properties

Label 40.72.1.bp.1
Level $40$
Index $72$
Genus $1$
Analytic rank $1$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $1600$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.74

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&3\\32&29\end{bmatrix}$, $\begin{bmatrix}25&24\\24&15\end{bmatrix}$, $\begin{bmatrix}25&29\\38&31\end{bmatrix}$, $\begin{bmatrix}27&10\\34&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.144.1-40.bp.1.1, 80.144.1-40.bp.1.2, 80.144.1-40.bp.1.3, 80.144.1-40.bp.1.4, 80.144.1-40.bp.1.5, 80.144.1-40.bp.1.6, 80.144.1-40.bp.1.7, 80.144.1-40.bp.1.8, 80.144.1-40.bp.1.9, 80.144.1-40.bp.1.10, 80.144.1-40.bp.1.11, 80.144.1-40.bp.1.12, 80.144.1-40.bp.1.13, 80.144.1-40.bp.1.14, 80.144.1-40.bp.1.15, 80.144.1-40.bp.1.16, 240.144.1-40.bp.1.1, 240.144.1-40.bp.1.2, 240.144.1-40.bp.1.3, 240.144.1-40.bp.1.4, 240.144.1-40.bp.1.5, 240.144.1-40.bp.1.6, 240.144.1-40.bp.1.7, 240.144.1-40.bp.1.8, 240.144.1-40.bp.1.9, 240.144.1-40.bp.1.10, 240.144.1-40.bp.1.11, 240.144.1-40.bp.1.12, 240.144.1-40.bp.1.13, 240.144.1-40.bp.1.14, 240.144.1-40.bp.1.15, 240.144.1-40.bp.1.16
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
Full 40-torsion field degree: $10240$

Jacobian

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

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 4 x y - x z - y^{2} - y z $
$=$ $5 x^{2} - 2 x y - 7 x z + 2 y^{2} + y z + 4 z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 29 x^{4} - 24 x^{3} z - 32 x^{2} y^{2} + 44 x^{2} z^{2} + 16 x y^{2} z - 24 x z^{3} - 2 y^{2} z^{2} + 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5}\cdot\frac{35916387258204054832500000000xz^{17}+1995301156576974724020600000000xz^{15}w^{2}+873129091676594052691260000000xz^{13}w^{4}-9842421414288768953122506000000xz^{11}w^{6}+22593902246399932583221155500000xz^{9}w^{8}-23077472986349986550314166460000xz^{7}w^{10}+8596406655566976850823224190000xz^{5}w^{12}+746752554867183290579014203400xz^{3}w^{14}+221468129207107403676956085xzw^{16}-1354469803119293790959100000000y^{2}z^{16}+1675541817554950514636280000000y^{2}z^{14}w^{2}+6400740707188889417195580000000y^{2}z^{12}w^{4}+10216700300125722603092047600000y^{2}z^{10}w^{6}-14472940311522239031180872900000y^{2}z^{8}w^{8}+5877729016747815852367930372000y^{2}z^{6}w^{10}+2474116210901072246478447649200y^{2}z^{4}w^{12}+71865937425924632405415169800y^{2}z^{2}w^{14}-9723435036901031279778233751y^{2}w^{16}+1817824165168179151473300000000yz^{17}+5434571129248415239088760000000yz^{15}w^{2}-7064074475627995132416900000000yz^{13}w^{4}-11820676864245537481024022800000yz^{11}w^{6}+5156879023296252139847674700000yz^{9}w^{8}+3648076798949471505284966084000yz^{7}w^{10}-4490324309183992249206790747600yz^{5}w^{12}+222773637825601266202159752200yz^{3}w^{14}-44638011132602855142180330147yzw^{16}-528176982931719354543600000000z^{18}-2542791376670255791400520000000z^{16}w^{2}-4510136454334910487967680000000z^{14}w^{4}+22868740289445321590258161600000z^{12}w^{6}-32914546976282322286163105200000z^{10}w^{8}+23125552024378994590039960392000z^{8}w^{10}-2888641763981829574942510300800z^{6}w^{12}-3465642187580809244374611821600z^{4}w^{14}-1786678955731759929626387916z^{2}w^{16}-500492947360694697575042w^{18}}{66511828255933434875000000xz^{17}+306789893134422059570000000xz^{15}w^{2}-1102857848318956371673250000xz^{13}w^{4}+409040359229342293636700000xz^{11}w^{6}+1325387434399007026261142500xz^{9}w^{8}-1296115304388668724078511000xz^{7}w^{10}+201245086822493484349911025xz^{5}w^{12}+89636186380801157359776910xz^{3}w^{14}+1000985894721389395150084xzw^{16}-2508277413183877390665000000y^{2}z^{16}+11682463291057145803458000000y^{2}z^{14}w^{2}-22907887285200051115839050000y^{2}z^{12}w^{4}+24790986939651661506588780000y^{2}z^{10}w^{6}-16194781515987907084502775500y^{2}z^{8}w^{8}+6378368220051420839434370600y^{2}z^{6}w^{10}-1314702441811170428129876155y^{2}z^{4}w^{12}+70065287830197650233114470y^{2}z^{2}w^{14}+4946284112529418945312500y^{2}w^{16}+3366341046607739169395000000yz^{17}-15069411586765687822254000000yz^{15}w^{2}+27681440359495628532795150000yz^{13}w^{4}-26256805246236025876735140000yz^{11}w^{6}+12821015451781458758221306500yz^{9}w^{8}-2187762246102102889092411800yz^{7}w^{10}-655930691683018393238770535yz^{5}w^{12}+349720241123353271649940590yz^{3}w^{14}-48703931932337787534334396yzw^{16}-978105523947628434340000000z^{18}+3457656100740936188068000000z^{16}w^{2}-4563164839584791801555800000z^{14}w^{4}+2680120946725003582150080000z^{12}w^{6}-753194380803993335321718000z^{10}w^{8}+304324751621549003001403600z^{8}w^{10}-150532665331311509171949380z^{6}w^{12}-17833220745241710526002200z^{4}w^{14}+19937289002242560197899120z^{2}w^{16}}$

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
20.36.0.b.2 $20$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.1.g.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
40.144.5.u.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.by.2 $40$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
40.144.5.da.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.df.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.in.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.it.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.ix.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.jg.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.ce.1 $40$ $5$ $5$ $13$ $3$ $1^{6}\cdot2^{3}$
120.144.5.chl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.chp.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cin.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cir.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eha.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eic.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eif.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.un.1 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.ier.1 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bp.1 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bge.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgf.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgs.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgt.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bou.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bov.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpi.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpj.1 $280$ $2$ $2$ $5$ $?$ not computed