Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $19$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $1 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 6 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $3$ are rational) | Cusp widths | $1^{3}\cdot19^{3}$ | Cusp orbits | $1^{3}\cdot3$ | ||
| Elliptic points: | $0$ of order $2$ and $6$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $3$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 19B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.120.1.2 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}15&4\\19&37\end{bmatrix}$, $\begin{bmatrix}31&25\\12&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 19.60.1.a.2 for the level structure with $-I$) |
| Cyclic 38-isogeny field degree: | $3$ |
| Cyclic 38-torsion field degree: | $18$ |
| Full 38-torsion field degree: | $6156$ |
Jacobian
| Conductor: | $19$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 19.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} + x^{2} + x $ |
Rational points
This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:-1:1)$, $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{25x^{2}y^{19}+437x^{2}y^{18}z-5377x^{2}y^{17}z^{2}-93708x^{2}y^{16}z^{3}-488965x^{2}y^{15}z^{4}-1364979x^{2}y^{14}z^{5}-2285282x^{2}y^{13}z^{6}-2008449x^{2}y^{12}z^{7}+542069x^{2}y^{11}z^{8}+4821139x^{2}y^{10}z^{9}+8563891x^{2}y^{9}z^{10}+9684705x^{2}y^{8}z^{11}+8068878x^{2}y^{7}z^{12}+5217654x^{2}y^{6}z^{13}+2701147x^{2}y^{5}z^{14}+1140921x^{2}y^{4}z^{15}+373267x^{2}y^{3}z^{16}+83743x^{2}y^{2}z^{17}+12875x^{2}yz^{18}+278x^{2}z^{19}-267xy^{19}z-4256xy^{18}z^{2}-9272xy^{17}z^{3}+101555xy^{16}z^{4}+807880xy^{15}z^{5}+3109277xy^{14}z^{6}+8115662xy^{13}z^{7}+15952598xy^{12}z^{8}+24810868xy^{11}z^{9}+31250103xy^{10}z^{10}+32107173xy^{9}z^{11}+26846076xy^{8}z^{12}+18062113xy^{7}z^{13}+9585355xy^{6}z^{14}+3950240xy^{5}z^{15}+1257209xy^{4}z^{16}+291645xy^{3}z^{17}+40578xy^{2}z^{18}+1392xyz^{19}-785xz^{20}-y^{21}-20y^{20}z+1671y^{19}z^{2}+26125y^{18}z^{3}+123481y^{17}z^{4}+177289y^{16}z^{5}-593997y^{15}z^{6}-3810505y^{14}z^{7}-10835971y^{13}z^{8}-20767518y^{12}z^{9}-29724629y^{11}z^{10}-33103707y^{10}z^{11}-29138305y^{9}z^{12}-20296063y^{8}z^{13}-11133608y^{7}z^{14}-4777770y^{6}z^{15}-1569899y^{5}z^{16}-370203y^{4}z^{17}-53403y^{3}z^{18}-1485y^{2}z^{19}+806yz^{20}+z^{21}}{z^{6}(58x^{2}y^{12}z+348x^{2}y^{11}z^{2}+1046x^{2}y^{10}z^{3}+2040x^{2}y^{9}z^{4}+3004x^{2}y^{8}z^{5}+3604x^{2}y^{7}z^{6}+3672x^{2}y^{6}z^{7}+3142x^{2}y^{5}z^{8}+2154x^{2}y^{4}z^{9}+1110x^{2}y^{3}z^{10}+393x^{2}y^{2}z^{11}+83x^{2}yz^{12}+7x^{2}z^{13}+xy^{14}+7xy^{13}z-67xy^{12}z^{2}-493xy^{11}z^{3}-1683xy^{10}z^{4}-3729xy^{9}z^{5}-5894xy^{8}z^{6}-6911xy^{7}z^{7}-5999xy^{6}z^{8}-3747xy^{5}z^{9}-1540xy^{4}z^{10}-303xy^{3}z^{11}+48xy^{2}z^{12}+42xyz^{13}+7xz^{14}-12y^{14}z-84y^{13}z^{2}-179y^{12}z^{3}+18y^{11}z^{4}+878y^{10}z^{5}+2223y^{9}z^{6}+3155y^{8}z^{7}+2912y^{7}z^{8}+1732y^{6}z^{9}+539y^{5}z^{10}-48y^{4}z^{11}-124y^{3}z^{12}-49y^{2}z^{13}-7yz^{14})}$ |
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 | $38$ | $3$ | $3$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 38.240.4-38.a.2.1 | $38$ | $2$ | $2$ | $4$ | $0$ | $1\cdot2$ |
| 38.240.4-38.c.2.2 | $38$ | $2$ | $2$ | $4$ | $1$ | $1\cdot2$ |
| 38.360.7-19.a.2.2 | $38$ | $3$ | $3$ | $7$ | $0$ | $6$ |
| 38.360.7-19.b.2.2 | $38$ | $3$ | $3$ | $7$ | $0$ | $6$ |
| 38.360.7-19.d.2.2 | $38$ | $3$ | $3$ | $7$ | $0$ | $6$ |
| 38.360.10-38.a.1.2 | $38$ | $3$ | $3$ | $10$ | $0$ | $1^{3}\cdot2\cdot4$ |
| 38.2280.64-19.a.1.1 | $38$ | $19$ | $19$ | $64$ | $8$ | $1^{3}\cdot2^{7}\cdot3^{2}\cdot4^{5}\cdot6^{2}\cdot8$ |
| 76.240.4-76.a.2.3 | $76$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 76.240.4-76.c.2.1 | $76$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 76.480.16-76.d.2.7 | $76$ | $4$ | $4$ | $16$ | $?$ | not computed |
| 114.240.4-114.a.2.1 | $114$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 114.240.4-114.c.2.2 | $114$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 114.360.13-57.a.2.8 | $114$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 114.480.13-57.a.1.7 | $114$ | $4$ | $4$ | $13$ | $?$ | not computed |
| 152.240.4-152.a.2.3 | $152$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 152.240.4-152.b.2.3 | $152$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 152.240.4-152.e.2.3 | $152$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 152.240.4-152.f.2.3 | $152$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 190.240.4-190.a.2.4 | $190$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 190.240.4-190.c.2.3 | $190$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.240.4-228.a.2.1 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 228.240.4-228.c.2.1 | $228$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 266.240.4-266.g.2.1 | $266$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 266.240.4-266.k.2.2 | $266$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 266.360.7-133.a.2.2 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 266.360.7-133.b.2.3 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 266.360.7-133.c.2.2 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 266.360.7-133.d.2.1 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 266.360.7-133.g.2.2 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 266.360.7-133.h.2.1 | $266$ | $3$ | $3$ | $7$ | $?$ | not computed |