Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse and Zureick-Brown (RZB) label: | X147 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.27 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&5\\0&1\end{bmatrix}$, $\begin{bmatrix}7&2\\6&1\end{bmatrix}$, $\begin{bmatrix}7&3\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.C_2^3$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $16$ |
| Full 8-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x - 14 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(-2:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{1404x^{2}y^{6}+3761589x^{2}y^{4}z^{2}+641729592x^{2}y^{2}z^{4}+19948109851x^{2}z^{6}+31626xy^{6}z+27474288xy^{4}z^{3}+3061839787xy^{2}z^{5}+76369887178xz^{7}+27y^{8}+410896y^{6}z^{2}+132195520y^{4}z^{4}+6979314188y^{2}z^{6}+72947334979z^{8}}{4x^{2}y^{6}-17x^{2}y^{4}z^{2}-8x^{2}y^{2}z^{4}+x^{2}z^{6}-2xy^{6}z+32xy^{4}z^{3}+17xy^{2}z^{5}-2xz^{7}+y^{8}-48y^{6}z^{2}+128y^{4}z^{4}+52y^{2}z^{6}-7z^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.s.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.48.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.v.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.ba.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bg.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.2.bc.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2.bd.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2.be.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2.bf.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 24.48.1.ga.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ge.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gq.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gu.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.em.1 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 24.96.5.cg.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.48.1.fc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fg.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fs.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fw.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.ca.1 | $40$ | $5$ | $5$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 40.144.9.do.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.nm.1 | $40$ | $10$ | $10$ | $17$ | $7$ | $1^{12}\cdot2^{2}$ |
| 48.48.2.u.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.v.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.w.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.x.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 56.48.1.fc.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fg.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fs.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fw.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.cg.1 | $56$ | $8$ | $8$ | $13$ | $4$ | $1^{12}$ |
| 56.504.37.em.1 | $56$ | $21$ | $21$ | $37$ | $17$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.em.1 | $56$ | $28$ | $28$ | $49$ | $21$ | $1^{20}\cdot2^{12}\cdot4$ |
| 80.48.2.ba.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bb.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bc.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bd.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 88.48.1.fc.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fg.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fs.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fw.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.cg.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.fc.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fg.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fs.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fw.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.2.u.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.v.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.w.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.x.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 120.48.1.ug.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.va.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fc.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fg.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fs.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fw.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fc.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fg.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fs.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fw.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ue.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ui.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.uu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.uy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.2.u.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.v.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.w.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.x.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 184.48.1.fc.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fg.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fs.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fw.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.2.ba.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bb.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bc.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bd.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 232.48.1.fc.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fg.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fs.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fw.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.2.ba.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bb.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bc.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bd.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 248.48.1.fc.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fg.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fs.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fw.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ue.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ui.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.uu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.uy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.2.ba.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bb.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bc.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bd.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 280.48.1.ti.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ty.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.uc.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fc.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fg.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fs.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fw.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.2.u.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.v.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.w.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.x.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 312.48.1.ug.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.va.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fc.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fg.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fs.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fw.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |