Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $38$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot19\cdot38$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 38A4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.120.4.10 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}9&31\\0&21\end{bmatrix}$, $\begin{bmatrix}31&8\\0&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 38.60.4.a.1 for the level structure with $-I$) |
| Cyclic 38-isogeny field degree: | $1$ |
| Cyclic 38-torsion field degree: | $18$ |
| Full 38-torsion field degree: | $6156$ |
Jacobian
| Conductor: | $2^{2}\cdot19^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 19.2.a.a$^{2}$, 38.2.a.a, 38.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} - x y + 2 x z + 2 x w - y z - y w + z w $ |
| $=$ | $ - x^{2} z + x^{2} w - x y w + x z^{2} + 2 x w^{2} + y^{3} + y z w + y w^{2} - z w^{2}$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:0:-1:1)$, $(-2:0:0:1)$, $(0:0:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{6201217392xy^{2}z^{7}+3435252496xyz^{8}-6506578672y^{2}z^{8}+36872304672xz^{9}-16351570160yz^{9}-16z^{10}-14880051216xy^{2}z^{6}w-7356905664xyz^{7}w+19177094064y^{2}z^{7}w-75788030096xz^{8}w+40396764832yz^{8}w+16351570064z^{9}w+23508635328xy^{2}z^{5}w^{2}+7406126304xyz^{6}w^{2}-32029169136y^{2}z^{6}w^{2}+111529182928xz^{7}w^{2}-61169759328yz^{7}w^{2}-50241768112z^{8}w^{2}-25115953440xy^{2}z^{4}w^{3}-15256636736xyz^{5}w^{3}+45665287520y^{2}z^{5}w^{3}-122836944768xz^{6}w^{3}+67119527200yz^{6}w^{3}+89896960672z^{7}w^{3}+18303371280xy^{2}z^{3}w^{4}+6680863392xyz^{4}w^{4}-44681163552y^{2}z^{4}w^{4}+129106140448xz^{5}w^{4}-73192979216yz^{5}w^{4}-117217257536z^{6}w^{4}-17391259584xy^{2}z^{2}w^{5}-8868322176xyz^{3}w^{5}+37462556976y^{2}z^{3}w^{5}-89298011840xz^{4}w^{5}+42316821888yz^{4}w^{5}+131713469552z^{5}w^{5}+6148767552xy^{2}zw^{6}+88770688xyz^{2}w^{6}-25617808096y^{2}z^{2}w^{6}+68950940736xz^{3}w^{6}-40687828448yz^{3}w^{6}-109275926480z^{4}w^{6}-3010936320xy^{2}w^{7}-4337971200xyzw^{7}+14041561920y^{2}zw^{7}-32072375552xz^{2}w^{7}+8245618816yz^{2}w^{7}+87776948416z^{3}w^{7}-6180720768y^{2}w^{8}+15054637312xzw^{8}-7939602048yzw^{8}-50899157632z^{2}w^{8}-6021872640xw^{9}+1221120000yw^{9}+22866371072zw^{9}-12043745536w^{10}}{-686969xy^{2}z^{7}-2645394xyz^{8}+934852y^{2}z^{8}+3292690xz^{9}-2050227yz^{9}-1080436xy^{2}z^{6}w-7970912xyz^{7}w+2762515y^{2}z^{7}w+8920802xz^{8}w-7725334yz^{8}w+2050227z^{9}w-699004xy^{2}z^{5}w^{2}-15706080xyz^{6}w^{2}+5933652y^{2}z^{6}w^{2}+18349550xz^{7}w^{2}-16697144yz^{7}w^{2}+4740239z^{8}w^{2}+381412xy^{2}z^{4}w^{3}-18152152xyz^{5}w^{3}+7499508y^{2}z^{5}w^{3}+21981064xz^{6}w^{3}-22452724yz^{6}w^{3}+9321398z^{7}w^{3}+988952xy^{2}z^{3}w^{4}-15844256xyz^{4}w^{4}+7386104y^{2}z^{4}w^{4}+19679036xz^{5}w^{4}-19731300yz^{5}w^{4}+8871024z^{6}w^{4}+616976xy^{2}z^{2}w^{5}-8719648xyz^{3}w^{5}+4781344y^{2}z^{3}w^{5}+10891792xz^{4}w^{5}-11696232yz^{4}w^{5}+6595496z^{5}w^{5}+308352xy^{2}zw^{6}-3606896xyz^{2}w^{6}+2223248y^{2}z^{2}w^{6}+4354400xz^{3}w^{6}-3607968yz^{3}w^{6}+1842160z^{4}w^{6}-589152xyzw^{7}+589152y^{2}zw^{7}+740320xz^{2}w^{7}-308688yz^{2}w^{7}+278288z^{3}w^{7}-19856y^{2}w^{8}-64xzw^{8}+674784yzw^{8}-502288z^{2}w^{8}+152640yw^{9}-152640zw^{9}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.40.1-19.a.1.1 | $38$ | $3$ | $3$ | $1$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.240.8-38.a.1.2 | $38$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 38.240.8-38.e.1.1 | $38$ | $2$ | $2$ | $8$ | $2$ | $1^{4}$ |
| 38.360.10-38.a.1.2 | $38$ | $3$ | $3$ | $10$ | $0$ | $2\cdot4$ |
| 38.360.10-38.a.2.3 | $38$ | $3$ | $3$ | $10$ | $0$ | $2\cdot4$ |
| 38.360.10-38.b.1.4 | $38$ | $3$ | $3$ | $10$ | $2$ | $1^{2}\cdot2^{2}$ |
| 38.2280.76-38.a.1.2 | $38$ | $19$ | $19$ | $76$ | $26$ | $1^{14}\cdot2^{12}\cdot3^{6}\cdot4^{4}$ |
| 76.240.8-76.a.1.3 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.8-76.b.1.7 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.8-76.c.1.11 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.8-76.d.1.1 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.8-76.e.1.7 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.8-76.f.1.7 | $76$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 76.240.9-76.a.1.7 | $76$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 76.240.9-76.b.1.7 | $76$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 76.240.9-76.c.1.7 | $76$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 76.240.9-76.d.1.11 | $76$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 114.240.8-114.a.1.1 | $114$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 114.240.8-114.c.1.2 | $114$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 114.360.14-114.a.1.14 | $114$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 114.480.17-114.a.1.15 | $114$ | $4$ | $4$ | $17$ | $?$ | not computed |
| 152.240.8-152.a.1.1 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.b.1.1 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.c.1.13 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.d.1.13 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.e.1.1 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.f.1.1 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.g.1.13 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.8-152.h.1.13 | $152$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 152.240.9-152.a.1.14 | $152$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 152.240.9-152.b.1.14 | $152$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 152.240.9-152.c.1.14 | $152$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 152.240.9-152.d.1.14 | $152$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 190.240.8-190.a.1.2 | $190$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 190.240.8-190.b.1.1 | $190$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.a.1.2 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.b.1.14 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.c.1.14 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.d.1.2 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.e.1.14 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.8-228.f.1.14 | $228$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 228.240.9-228.a.1.13 | $228$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 228.240.9-228.b.1.13 | $228$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 228.240.9-228.e.1.13 | $228$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 228.240.9-228.f.1.13 | $228$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 266.240.8-266.g.1.1 | $266$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 266.240.8-266.h.1.1 | $266$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 266.360.10-266.a.1.8 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 266.360.10-266.a.2.8 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 266.360.10-266.b.1.4 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 266.360.10-266.b.2.8 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 266.360.10-266.c.1.4 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 266.360.10-266.c.2.8 | $266$ | $3$ | $3$ | $10$ | $?$ | not computed |