Invariants
| Level: | $165$ | $\SL_2$-level: | $15$ | Newform level: | $15$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot3\cdot5\cdot15$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 15C1 |
Level structure
| $\GL_2(\Z/165\Z)$-generators: | $\begin{bmatrix}25&111\\52&89\end{bmatrix}$, $\begin{bmatrix}81&56\\94&38\end{bmatrix}$, $\begin{bmatrix}116&55\\147&49\end{bmatrix}$, $\begin{bmatrix}162&34\\64&117\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 15.24.1.a.1 for the level structure with $-I$) |
| Cyclic 165-isogeny field degree: | $12$ |
| Cyclic 165-torsion field degree: | $960$ |
| Full 165-torsion field degree: | $6336000$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 15.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + \left(x + 1\right) y $ | $=$ | $ x^{3} + x^{2} - 10x - 10 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{7x^{2}y^{8}-168x^{2}y^{7}z+8649x^{2}y^{6}z^{2}+71386x^{2}y^{5}z^{3}-1633410x^{2}y^{4}z^{4}+47996469x^{2}y^{3}z^{5}+121210709x^{2}y^{2}z^{6}+524262998x^{2}yz^{7}+3323839242x^{2}z^{8}+xy^{9}-79xy^{8}z-2443xy^{7}z^{2}+12013xy^{6}z^{3}-2439265xy^{5}z^{4}-13029558xy^{4}z^{5}-137755648xy^{3}z^{6}-1035181711xy^{2}z^{7}+543404161xyz^{8}+13648916130xz^{9}-582y^{8}z^{2}+34523y^{7}z^{3}-9219y^{6}z^{4}+6428904y^{5}z^{5}+108506910y^{4}z^{6}-371358519y^{3}z^{7}-3090420549y^{2}z^{8}-3467643238yz^{9}+10325076888z^{10}}{z^{2}(12x^{2}y^{6}-189x^{2}y^{5}z-900x^{2}y^{4}z^{2}+2847x^{2}y^{3}z^{3}+61647x^{2}y^{2}z^{4}+260822x^{2}yz^{5}+1943616x^{2}z^{6}-4xy^{7}+7xy^{6}z+412xy^{5}z^{2}-2863xy^{4}z^{3}-11906xy^{3}z^{4}+337150xy^{2}z^{5}+1639558xyz^{6}+8447100xz^{7}-y^{8}+46y^{7}z-21y^{6}z^{2}-2147y^{5}z^{3}+2767y^{4}z^{4}+53165y^{3}z^{5}+782155y^{2}z^{6}+1378736yz^{7}+6503484z^{8})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
| 33.8.0-3.a.1.1 | $33$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 33.8.0-3.a.1.1 | $33$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 165.96.1-15.a.1.7 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-15.a.2.7 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-165.a.1.3 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-165.a.2.4 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-15.b.1.4 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-15.b.2.4 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-165.b.1.8 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.96.1-165.b.2.6 | $165$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 165.144.3-15.a.1.1 | $165$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 165.240.5-15.a.1.5 | $165$ | $5$ | $5$ | $5$ | $?$ | not computed |
| 330.96.3-30.a.1.7 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.a.1.5 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.b.1.5 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.b.1.4 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.c.1.5 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.c.1.7 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.d.1.5 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.d.1.13 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.e.1.5 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.e.2.3 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.e.1.8 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.e.2.8 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.f.1.2 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-30.f.2.2 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.f.1.14 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.96.3-330.f.2.14 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.144.3-30.a.1.5 | $330$ | $3$ | $3$ | $3$ | $?$ | not computed |