Invariants
| Level: | $15$ | $\SL_2$-level: | $15$ | Newform level: | $225$ | ||
| Index: | $15$ | $\PSL_2$-index: | $15$ | ||||
| Genus: | $1 = 1 + \frac{ 15 }{12} - \frac{ 3 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $15$ | Cusp orbits | $1$ | ||
| Elliptic points: | $3$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-11,-16,-19$) |
Other labels
| Cummins and Pauli (CP) label: | 15A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 15.15.1.1 |
Level structure
| $\GL_2(\Z/15\Z)$-generators: | $\begin{bmatrix}0&1\\13&0\end{bmatrix}$, $\begin{bmatrix}2&6\\9&8\end{bmatrix}$, $\begin{bmatrix}13&2\\13&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 15-isogeny field degree: | $24$ |
| Cyclic 15-torsion field degree: | $192$ |
| Full 15-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 225.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} - 34 $ |
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 15 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(y+3z)^{3}(y^{2}+yz+34z^{2})}{z^{5}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(5)$ | $5$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(5)$ | $5$ | $3$ | $3$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 15.30.2.a.1 | $15$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 15.30.2.b.1 | $15$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 15.30.2.c.1 | $15$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 15.30.2.d.1 | $15$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 15.45.1.a.1 | $15$ | $3$ | $3$ | $1$ | $1$ | dimension zero |
| 15.60.4.e.1 | $15$ | $4$ | $4$ | $4$ | $2$ | $1\cdot2$ |
| 30.30.2.e.1 | $30$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 30.30.2.f.1 | $30$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 30.30.2.g.1 | $30$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 30.30.2.h.1 | $30$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 30.30.3.a.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.30.3.b.1 | $30$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 30.30.3.c.1 | $30$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 30.30.3.d.1 | $30$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 30.45.3.a.1 | $30$ | $3$ | $3$ | $3$ | $2$ | $1^{2}$ |
| 45.45.3.a.1 | $45$ | $3$ | $3$ | $3$ | $3$ | $2$ |
| 45.135.9.a.1 | $45$ | $9$ | $9$ | $9$ | $9$ | $2^{2}\cdot4$ |
| 60.30.2.k.1 | $60$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 60.30.2.l.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.30.2.o.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.30.2.p.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.30.2.q.1 | $60$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 60.30.2.r.1 | $60$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 60.30.2.s.1 | $60$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 60.30.2.t.1 | $60$ | $2$ | $2$ | $2$ | $2$ | $1$ |
| 60.30.3.a.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.30.3.b.1 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.30.3.c.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.30.3.d.1 | $60$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 60.60.4.cq.1 | $60$ | $4$ | $4$ | $4$ | $4$ | $1^{3}$ |
| 105.30.2.a.1 | $105$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 105.30.2.b.1 | $105$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 105.30.2.c.1 | $105$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 105.30.2.d.1 | $105$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 105.120.10.a.1 | $105$ | $8$ | $8$ | $10$ | $?$ | not computed |
| 105.315.22.a.1 | $105$ | $21$ | $21$ | $22$ | $?$ | not computed |
| 120.30.2.o.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.p.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.q.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.r.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.u.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.v.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.w.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.x.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.y.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.z.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.ba.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.bb.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.bc.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.bd.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.be.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.2.bf.1 | $120$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.30.3.a.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.b.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.c.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.d.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.e.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.f.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.g.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.30.3.h.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 165.30.2.a.1 | $165$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 165.30.2.b.1 | $165$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 165.30.2.c.1 | $165$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 165.30.2.d.1 | $165$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 165.180.15.a.1 | $165$ | $12$ | $12$ | $15$ | $?$ | not computed |
| 195.30.2.a.1 | $195$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 195.30.2.b.1 | $195$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 195.30.2.c.1 | $195$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 195.30.2.d.1 | $195$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 195.210.16.a.1 | $195$ | $14$ | $14$ | $16$ | $?$ | not computed |
| 210.30.2.e.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.30.2.f.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.30.2.g.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.30.2.h.1 | $210$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 210.30.3.a.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.30.3.b.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.30.3.c.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 210.30.3.d.1 | $210$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 255.30.2.a.1 | $255$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 255.30.2.b.1 | $255$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 255.30.2.c.1 | $255$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 255.30.2.d.1 | $255$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 255.270.21.a.1 | $255$ | $18$ | $18$ | $21$ | $?$ | not computed |
| 285.30.2.a.1 | $285$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 285.30.2.b.1 | $285$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 285.30.2.c.1 | $285$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 285.30.2.d.1 | $285$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 330.30.2.e.1 | $330$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 330.30.2.f.1 | $330$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 330.30.2.g.1 | $330$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 330.30.2.h.1 | $330$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 330.30.3.a.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.30.3.b.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.30.3.c.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 330.30.3.d.1 | $330$ | $2$ | $2$ | $3$ | $?$ | not computed |