Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.43 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&10\\12&17\end{bmatrix}$, $\begin{bmatrix}7&16\\0&13\end{bmatrix}$, $\begin{bmatrix}7&20\\12&11\end{bmatrix}$, $\begin{bmatrix}11&6\\12&19\end{bmatrix}$, $\begin{bmatrix}19&0\\0&1\end{bmatrix}$, $\begin{bmatrix}19&14\\12&19\end{bmatrix}$, $\begin{bmatrix}19&20\\12&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^6$ |
| Contains $-I$: | no $\quad$ (see 12.48.1.b.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} - 4x + 4 $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(1:0:1)$, $(0:-2:1)$, $(-2:0:1)$, $(4:6:1)$, $(2:0:1)$, $(4:-6:1)$, $(0:1:0)$, $(0:2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{16x^{2}y^{14}-254370x^{2}y^{12}z^{2}+272792952x^{2}y^{10}z^{4}-43885840685x^{2}y^{8}z^{6}+1881465528280x^{2}y^{6}z^{8}-31188954184569x^{2}y^{4}z^{10}+218165771777620x^{2}y^{2}z^{12}-541652977285693x^{2}z^{14}-852xy^{14}z+2923620xy^{12}z^{3}-2032924815xy^{10}z^{5}+217907337732xy^{8}z^{7}-7546917207824xy^{6}z^{9}+109553117422806xy^{4}z^{11}-699635304814401xy^{2}z^{13}+1624959302500352xz^{15}-y^{16}+11420y^{14}z^{2}-33883710y^{12}z^{4}+9961460717y^{10}z^{6}-628484953611y^{8}z^{8}+14424778023672y^{6}z^{10}-146070717545179y^{4}z^{12}+662020649013103y^{2}z^{14}-1083306712635148z^{16}}{zy^{4}(77x^{2}y^{8}z+128x^{2}y^{6}z^{3}-1280x^{2}y^{4}z^{5}+2048x^{2}y^{2}z^{7}+16384x^{2}z^{9}+xy^{10}-224xy^{8}z^{2}+384xy^{6}z^{4}-2048xy^{4}z^{6}+6144xy^{2}z^{8}+16384xz^{10}-15y^{10}z-20y^{8}z^{3}+1280y^{4}z^{7}+8192y^{2}z^{9}-32768z^{11})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-4.b.1.6 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-4.b.1.6 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.48.0-6.a.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-6.a.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.1-12.b.1.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-12.b.2.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-12.b.3.17 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-12.b.4.17 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ck.1.25 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ck.2.25 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ck.3.23 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ck.4.23 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-12.b.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-12.e.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-12.e.1.26 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.f.1.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.f.1.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-12.g.1.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-12.g.1.28 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-12.g.2.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-12.g.2.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bf.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.bf.1.17 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.bk.1.32 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bk.2.32 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bl.1.36 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bl.2.36 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.1.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.1.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.2.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.2.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.3.43 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.3.59 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.4.45 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bm.4.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bn.1.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bn.1.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bn.2.57 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bn.2.59 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bo.1.45 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bo.1.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bo.2.51 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bo.2.55 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bp.1.43 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bp.1.59 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bp.2.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bp.2.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bq.1.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bq.1.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bq.2.57 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bq.2.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.1.57 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.1.59 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.2.57 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.2.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.3.53 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.3.55 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.4.57 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.4.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bv.1.22 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bv.1.24 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bv.2.20 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bv.2.24 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.cg.1.12 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ch.1.19 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ck.1.32 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.cl.1.33 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.5-24.c.1.39 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.d.1.15 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.g.1.38 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.h.1.35 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.192.5-24.k.1.35 | $24$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 24.192.5-24.k.2.26 | $24$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 24.192.5-24.l.1.34 | $24$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 24.192.5-24.l.2.21 | $24$ | $2$ | $2$ | $5$ | $0$ | $4$ |
| 24.288.5-12.b.1.26 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 72.288.5-36.b.1.34 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-36.b.1.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-36.f.1.16 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-60.b.1.36 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.2.35 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.3.35 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-60.b.4.35 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.1.43 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.2.48 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.3.47 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.lg.4.41 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-60.c.1.44 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.c.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.f.1.34 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.f.1.60 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.1.13 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.2.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-60.l.2.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ds.1.21 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ds.1.61 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dz.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.dz.1.73 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eq.1.54 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eq.2.73 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.er.1.60 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.er.2.65 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.1.47 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.1.92 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.2.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.2.120 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.3.60 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.3.87 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.4.43 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.es.4.120 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.1.29 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.1.123 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.2.57 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.et.2.120 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.1.61 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.1.115 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.2.25 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.eu.2.128 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.1.63 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.1.89 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.2.45 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ev.2.96 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.1.109 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.2.60 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ew.2.93 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.1.107 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.2.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.2.123 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.3.56 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.3.91 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.4.40 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ex.4.123 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.1.46 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.2.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fb.2.46 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fy.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.fz.1.41 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gc.1.52 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.gd.1.43 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.c.1.34 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.d.1.28 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.g.1.42 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.h.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.k.1.24 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.k.2.40 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.l.1.41 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.l.2.51 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.480.17-60.b.1.54 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-84.b.1.34 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-84.b.2.33 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-84.b.3.33 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-84.b.4.35 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lg.1.46 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lg.2.46 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lg.3.46 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.lg.4.46 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-84.c.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-84.c.1.50 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-84.f.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
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| 312.192.5-312.h.1.23 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.k.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.k.2.64 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.l.1.35 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.l.2.70 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |