Invariants
| Level: | $120$ | $\SL_2$-level: | $20$ | Newform level: | $20$ | ||
| Index: | $144$ | $\PSL_2$-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 | $2^{6}\cdot10^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 72$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 10K1 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}29&46\\92&35\end{bmatrix}$, $\begin{bmatrix}39&26\\2&65\end{bmatrix}$, $\begin{bmatrix}41&84\\12&49\end{bmatrix}$, $\begin{bmatrix}77&106\\102&83\end{bmatrix}$, $\begin{bmatrix}89&28\\64&115\end{bmatrix}$, $\begin{bmatrix}119&80\\14&103\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.72.1.b.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $16$ |
| Cyclic 120-torsion field degree: | $512$ |
| Full 120-torsion field degree: | $245760$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 20.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + 2 x z - x w - z^{2} + 2 z w $ |
| $=$ | $5 x^{2} - 15 x y - 3 x z + 2 x w + 15 y^{2} + 2 z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 3 x^{3} y - 5 x^{3} z + 3 x^{2} y^{2} + 12 x^{2} y z + 6 x^{2} z^{2} - 12 x y^{2} z + \cdots + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
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{1}{3^5\cdot5}\cdot\frac{72088571495331161625274xz^{17}-596906015130411890052217xz^{16}w+2128881901190414121807184xz^{15}w^{2}-4212475344270316706979880xz^{14}w^{3}+4859546167776656344152220xz^{13}w^{4}-2875371838747922073470638xz^{12}w^{5}-15278070769469588481176xz^{11}w^{6}+1308482413765763871102512xz^{10}w^{7}-839521513387395133268210xz^{9}w^{8}+94847478086475452563765xz^{8}w^{9}+133077856290479757331952xz^{7}w^{10}-66015861787913010287876xz^{6}w^{11}+5836892144705246834522xz^{5}w^{12}+4028621104939591332295xz^{4}w^{13}-1388610190408138731580xz^{3}w^{14}+176404062445248495364xz^{2}w^{15}-9447585389455259542xzw^{16}+175030242880182319xw^{17}-24029575337157852046601z^{18}+229005762593808122517890z^{17}w-941442844980268311095612z^{16}w^{2}+2166077127675194042324192z^{15}w^{3}-2982597325179871141404170z^{14}w^{4}+2319968795231049227233172z^{13}w^{5}-581872215679893275555810z^{12}w^{6}-655302566808242300243176z^{11}w^{7}+685066630965131509058761z^{10}w^{8}-198532529800586315323490z^{9}w^{9}-64302321148927617767578z^{8}w^{10}+61679316780577916861560z^{7}w^{11}-12655375831780630407991z^{6}w^{12}-2313148026269317589954z^{5}w^{13}+1482257186461178620055z^{4}w^{14}-247812018165633270476z^{3}w^{15}+16270897770204054635z^{2}w^{16}-350070007006302302zw^{17}-528958107648w^{18}}{29192925742915376xz^{17}-361262455301859245xz^{16}w+2025943423623808430xz^{15}w^{2}-6821423104405060610xz^{14}w^{3}+15405892004394921470xz^{13}w^{4}-24702130952039547092xz^{12}w^{5}+29039869147729076210xz^{11}w^{6}-25495083993194401265xz^{10}w^{7}+16872947929951082780xz^{9}w^{8}-8439246973171210885xz^{8}w^{9}+3178991904203676538xz^{7}w^{10}-893231898608644420xz^{6}w^{11}+184064314895544430xz^{5}w^{12}-27080986234065610xz^{4}w^{13}+2728186707805270xz^{3}w^{14}-175782455755369xz^{2}w^{15}+6396484375000xzw^{16}-97656250000xw^{17}-9730975624272124z^{18}+132584543506188598z^{17}w-818998448574894490z^{16}w^{2}+3039266919432896290z^{15}w^{3}-7570372825611988495z^{14}w^{4}+13397903073412617928z^{13}w^{5}-17399506535464239781z^{12}w^{6}+16890443510158411870z^{11}w^{7}-12372731701066892170z^{10}w^{8}+6857746906611787310z^{9}w^{9}-2866674722269615772z^{8}w^{10}+895374745143327494z^{7}w^{11}-205530048072213515z^{6}w^{12}+33772063950605540z^{5}w^{13}-3811505581765445z^{4}w^{14}+276074848196426z^{3}w^{15}-11329757586752z^{2}w^{16}+195312500000zw^{17}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 30.72.1.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}-3X^{3}Y+3X^{2}Y^{2}-5X^{3}Z+12X^{2}YZ-12XY^{2}Z+6X^{2}Z^{2}-12XYZ^{2}+12Y^{2}Z^{2}-2XZ^{3}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 40.72.1-10.a.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 120.72.1-10.a.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 120.288.5-60.bc.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bc.1.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bd.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bd.1.24 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bd.1.35 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bg.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bg.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bg.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bh.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bh.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bh.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bo.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bo.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bo.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bp.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bp.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bp.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bs.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bs.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bs.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bt.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bt.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-60.bt.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dg.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dg.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dg.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dj.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dj.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dj.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ds.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ds.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ds.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dv.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dv.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dv.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eq.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eq.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eq.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.et.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.et.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.et.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fc.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fc.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.fc.1.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ff.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ff.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ff.1.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.432.13-30.b.1.1 | $120$ | $3$ | $3$ | $13$ | $?$ | not computed |