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Properties

Label 88.144.4-44.c.1.24
Level $88$
Index $144$
Genus $4$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $88$ $\SL_2$-level: $88$ Newform level: $44$
Index: $144$ $\PSL_2$-index:$72$
Genus: $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $1^{2}\cdot4\cdot11^{2}\cdot44$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 44D4

Level structure

$\GL_2(\Z/88\Z)$-generators: $\begin{bmatrix}9&86\\40&11\end{bmatrix}$, $\begin{bmatrix}20&9\\11&18\end{bmatrix}$, $\begin{bmatrix}49&60\\76&33\end{bmatrix}$, $\begin{bmatrix}50&69\\13&62\end{bmatrix}$, $\begin{bmatrix}51&74\\2&35\end{bmatrix}$, $\begin{bmatrix}59&30\\32&57\end{bmatrix}$
Contains $-I$: no $\quad$ (see 44.72.4.c.1 for the level structure with $-I$)
Cyclic 88-isogeny field degree: $2$
Cyclic 88-torsion field degree: $40$
Full 88-torsion field degree: $140800$

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ x^{2} + 2 x z - 2 y^{2} - 5 y z + y w + z w $
$=$ $x^{3} + 3 x^{2} z + 6 x z^{2} + y^{2} z + y z^{2} + y z w + 3 z^{2} w - z w^{2}$
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:1/2:0:1)$, $(0:1/3:1/6:1)$, $(0:0:1:0)$, $(0:-1:1/2:1)$, $(0:0:0:1)$, $(0:1:-1/4:1)$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{4607137258680y^{3}z^{9}+26539165354467xyz^{10}+73113181906963y^{2}z^{10}+80911051981644xz^{11}+241128258892210yz^{11}+237774892433408z^{12}-40089509424984y^{3}z^{8}w-81620437710171xyz^{9}w-296714492150956y^{2}z^{9}w-157324718028871xz^{10}w-776854963661124yz^{10}w-672869151309402z^{11}w+71256145055472y^{3}z^{7}w^{2}+104777506064772xyz^{8}w^{2}+440883840950834y^{2}z^{8}w^{2}+197020550333546xz^{9}w^{2}+1059208607876308yz^{9}w^{2}+929142878049447z^{10}w^{2}-54408959911248y^{3}z^{6}w^{3}-72503218104540xyz^{7}w^{3}-343280332252416y^{2}z^{7}w^{3}-143454998820684xz^{8}w^{3}-835176375728368yz^{8}w^{3}-770517000559872z^{9}w^{3}+24072647923392y^{3}z^{5}w^{4}+29297871576996xyz^{6}w^{4}+169324521148484y^{2}z^{6}w^{4}+67826300267648xz^{7}w^{4}+431999696980256yz^{7}w^{4}+425771099415524z^{8}w^{4}-5940238814784y^{3}z^{4}w^{5}-6929242334388xyz^{5}w^{5}-50002593690880y^{2}z^{5}w^{5}-19502225385076xz^{6}w^{5}-144783998201144yz^{6}w^{5}-160026619300064z^{7}w^{5}+616223754048y^{3}z^{3}w^{6}+482139764400xyz^{4}w^{6}+8024814772896y^{2}z^{4}w^{6}+2620206193104xz^{5}w^{6}+28940165866496yz^{5}w^{6}+39443292361820z^{6}w^{6}-126152778432y^{3}z^{2}w^{7}-31238362704xyz^{3}w^{7}-767544853696y^{2}z^{3}w^{7}-9807462352xz^{4}w^{7}-3259621969600yz^{4}w^{7}-6150493563328z^{5}w^{7}+14295305088y^{3}zw^{8}-14114529936xyz^{2}w^{8}+117067597904y^{2}z^{2}w^{8}-41590183840xz^{3}w^{8}+190296300640yz^{3}w^{8}+586487560336z^{4}w^{8}-1588367232y^{3}w^{9}+1568281104xyzw^{9}-13280390976y^{2}zw^{9}+9833878224xz^{2}w^{9}-21704211520yz^{2}w^{9}-29158602656z^{3}w^{9}+851528928y^{2}w^{10}-826559712xzw^{10}+1707383232yzw^{10}+2771937584z^{2}w^{10}-393216yw^{11}-786432zw^{11}+32768w^{12}}{17785290752y^{3}z^{9}-17267195904xyz^{10}+98631942144y^{2}z^{10}-16495017984xz^{11}+125976641536yz^{11}-24904859648y^{3}z^{8}w+29878616064xyz^{9}w-147519307776y^{2}z^{9}w+45040959488xz^{10}w-207406170112yz^{10}w-8247508992z^{11}w+10874388480y^{3}z^{7}w^{2}-21320990720xyz^{8}w^{2}+92307062784y^{2}z^{8}w^{2}-42044620800xz^{9}w^{2}+166278594560yz^{9}w^{2}+15162408960z^{10}w^{2}-3912105984y^{3}z^{6}w^{3}+11193122816xyz^{7}w^{3}-36189110272y^{2}z^{7}w^{3}+26939981824xz^{8}w^{3}-77490290688yz^{8}w^{3}-11588337664z^{9}w^{3}+149422080y^{3}z^{5}w^{4}-3203006464xyz^{6}w^{4}+7958429696y^{2}z^{6}w^{4}-10485923840xz^{7}w^{4}+25938198528yz^{7}w^{4}+7248052224z^{8}w^{4}+36175872y^{3}z^{4}w^{5}+777125888xyz^{5}w^{5}-635699200y^{2}z^{5}w^{5}+2910781440xz^{6}w^{5}-4866801664yz^{6}w^{5}-3127214080z^{7}w^{5}-33030144y^{3}z^{3}w^{6}-66060288xyz^{4}w^{6}-53346304y^{2}z^{4}w^{6}-525041664xz^{5}w^{6}+674562048yz^{5}w^{6}+877199360z^{6}w^{6}+4718592y^{3}z^{2}w^{7}+7864320xyz^{3}w^{7}+47710208y^{2}z^{3}w^{7}+50593792xz^{4}w^{7}-5963776yz^{4}w^{7}-215941120z^{5}w^{7}+589824xyz^{2}w^{8}-4194304y^{2}z^{2}w^{8}-3768320xz^{3}w^{8}-12222464yz^{3}w^{8}+20709376z^{4}w^{8}-65536xyzw^{9}-393216xz^{2}w^{9}+425984yz^{2}w^{9}-2195456z^{3}w^{9}+32768xzw^{10}+262144z^{2}w^{10}}$

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
8.12.0-4.c.1.5 $8$ $12$ $12$ $0$ $0$
$X_0(11)$ $11$ $12$ $6$ $1$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.12.0-4.c.1.5 $8$ $12$ $12$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
88.288.8-44.c.1.17 $88$ $2$ $2$ $8$
88.288.8-44.c.1.18 $88$ $2$ $2$ $8$
88.288.8-44.c.2.17 $88$ $2$ $2$ $8$
88.288.8-44.c.2.18 $88$ $2$ $2$ $8$
88.288.8-44.c.3.21 $88$ $2$ $2$ $8$
88.288.8-44.c.3.23 $88$ $2$ $2$ $8$
88.288.8-44.c.4.21 $88$ $2$ $2$ $8$
88.288.8-44.c.4.22 $88$ $2$ $2$ $8$
88.288.8-88.c.1.18 $88$ $2$ $2$ $8$
88.288.8-88.c.1.20 $88$ $2$ $2$ $8$
88.288.8-88.c.2.19 $88$ $2$ $2$ $8$
88.288.8-88.c.2.20 $88$ $2$ $2$ $8$
88.288.8-88.c.3.19 $88$ $2$ $2$ $8$
88.288.8-88.c.3.23 $88$ $2$ $2$ $8$
88.288.8-88.c.4.21 $88$ $2$ $2$ $8$
88.288.8-88.c.4.22 $88$ $2$ $2$ $8$
88.288.8-88.f.1.4 $88$ $2$ $2$ $8$
88.288.8-88.f.1.8 $88$ $2$ $2$ $8$
88.288.8-88.f.2.7 $88$ $2$ $2$ $8$
88.288.8-88.f.2.15 $88$ $2$ $2$ $8$
88.288.8-88.g.1.6 $88$ $2$ $2$ $8$
88.288.8-88.g.1.8 $88$ $2$ $2$ $8$
88.288.8-88.g.2.11 $88$ $2$ $2$ $8$
88.288.8-88.g.2.15 $88$ $2$ $2$ $8$
88.288.9-44.b.1.28 $88$ $2$ $2$ $9$
88.288.9-88.f.1.20 $88$ $2$ $2$ $9$
88.288.9-44.h.1.12 $88$ $2$ $2$ $9$
88.288.9-44.k.1.13 $88$ $2$ $2$ $9$
88.288.9-44.l.1.13 $88$ $2$ $2$ $9$
88.288.9-88.w.1.20 $88$ $2$ $2$ $9$
88.288.9-88.be.1.18 $88$ $2$ $2$ $9$
88.288.9-88.bh.1.18 $88$ $2$ $2$ $9$
88.288.9-88.bk.1.5 $88$ $2$ $2$ $9$
88.288.9-88.bk.1.13 $88$ $2$ $2$ $9$
88.288.9-88.bl.1.2 $88$ $2$ $2$ $9$
88.288.9-88.bl.1.10 $88$ $2$ $2$ $9$
88.288.9-88.bm.1.3 $88$ $2$ $2$ $9$
88.288.9-88.bm.1.7 $88$ $2$ $2$ $9$
88.288.9-88.bn.1.2 $88$ $2$ $2$ $9$
88.288.9-88.bn.1.6 $88$ $2$ $2$ $9$
88.288.9-88.bo.1.2 $88$ $2$ $2$ $9$
88.288.9-88.bo.1.6 $88$ $2$ $2$ $9$
88.288.9-88.bp.1.3 $88$ $2$ $2$ $9$
88.288.9-88.bp.1.7 $88$ $2$ $2$ $9$
88.288.9-88.bq.1.2 $88$ $2$ $2$ $9$
88.288.9-88.bq.1.10 $88$ $2$ $2$ $9$
88.288.9-88.br.1.5 $88$ $2$ $2$ $9$
88.288.9-88.br.1.13 $88$ $2$ $2$ $9$
88.288.10-88.e.1.2 $88$ $2$ $2$ $10$
88.288.10-88.e.1.4 $88$ $2$ $2$ $10$
88.288.10-88.e.2.3 $88$ $2$ $2$ $10$
88.288.10-88.e.2.7 $88$ $2$ $2$ $10$
88.288.10-88.f.1.2 $88$ $2$ $2$ $10$
88.288.10-88.f.1.4 $88$ $2$ $2$ $10$
88.288.10-88.f.2.3 $88$ $2$ $2$ $10$
88.288.10-88.f.2.7 $88$ $2$ $2$ $10$
264.288.8-132.c.1.26 $264$ $2$ $2$ $8$
264.288.8-132.c.1.38 $264$ $2$ $2$ $8$
264.288.8-132.c.2.26 $264$ $2$ $2$ $8$
264.288.8-132.c.2.38 $264$ $2$ $2$ $8$
264.288.8-132.c.3.27 $264$ $2$ $2$ $8$
264.288.8-132.c.3.29 $264$ $2$ $2$ $8$
264.288.8-132.c.4.27 $264$ $2$ $2$ $8$
264.288.8-132.c.4.29 $264$ $2$ $2$ $8$
264.288.8-264.ti.1.9 $264$ $2$ $2$ $8$
264.288.8-264.ti.1.13 $264$ $2$ $2$ $8$
264.288.8-264.ti.2.9 $264$ $2$ $2$ $8$
264.288.8-264.ti.2.13 $264$ $2$ $2$ $8$
264.288.8-264.ti.3.9 $264$ $2$ $2$ $8$
264.288.8-264.ti.3.10 $264$ $2$ $2$ $8$
264.288.8-264.ti.4.9 $264$ $2$ $2$ $8$
264.288.8-264.ti.4.10 $264$ $2$ $2$ $8$
264.288.8-264.tl.1.10 $264$ $2$ $2$ $8$
264.288.8-264.tl.1.14 $264$ $2$ $2$ $8$
264.288.8-264.tl.2.10 $264$ $2$ $2$ $8$
264.288.8-264.tl.2.14 $264$ $2$ $2$ $8$
264.288.8-264.tm.1.34 $264$ $2$ $2$ $8$
264.288.8-264.tm.1.38 $264$ $2$ $2$ $8$
264.288.8-264.tm.2.34 $264$ $2$ $2$ $8$
264.288.8-264.tm.2.38 $264$ $2$ $2$ $8$
264.288.9-132.k.1.24 $264$ $2$ $2$ $9$
264.288.9-132.l.1.24 $264$ $2$ $2$ $9$
264.288.9-132.o.1.24 $264$ $2$ $2$ $9$
264.288.9-132.p.1.24 $264$ $2$ $2$ $9$
264.288.9-264.ije.1.39 $264$ $2$ $2$ $9$
264.288.9-264.ijh.1.39 $264$ $2$ $2$ $9$
264.288.9-264.ijq.1.33 $264$ $2$ $2$ $9$
264.288.9-264.ijt.1.33 $264$ $2$ $2$ $9$
264.288.9-264.ijw.1.17 $264$ $2$ $2$ $9$
264.288.9-264.ijw.1.25 $264$ $2$ $2$ $9$
264.288.9-264.ijx.1.2 $264$ $2$ $2$ $9$
264.288.9-264.ijx.1.18 $264$ $2$ $2$ $9$
264.288.9-264.ijy.1.17 $264$ $2$ $2$ $9$
264.288.9-264.ijy.1.25 $264$ $2$ $2$ $9$
264.288.9-264.ijz.1.2 $264$ $2$ $2$ $9$
264.288.9-264.ijz.1.18 $264$ $2$ $2$ $9$
264.288.9-264.ika.1.2 $264$ $2$ $2$ $9$
264.288.9-264.ika.1.18 $264$ $2$ $2$ $9$
264.288.9-264.ikb.1.17 $264$ $2$ $2$ $9$
264.288.9-264.ikb.1.25 $264$ $2$ $2$ $9$
264.288.9-264.ikc.1.2 $264$ $2$ $2$ $9$
264.288.9-264.ikc.1.18 $264$ $2$ $2$ $9$
264.288.9-264.ikd.1.17 $264$ $2$ $2$ $9$
264.288.9-264.ikd.1.25 $264$ $2$ $2$ $9$
264.288.10-264.i.1.33 $264$ $2$ $2$ $10$
264.288.10-264.i.1.49 $264$ $2$ $2$ $10$
264.288.10-264.i.2.33 $264$ $2$ $2$ $10$
264.288.10-264.i.2.49 $264$ $2$ $2$ $10$
264.288.10-264.j.1.5 $264$ $2$ $2$ $10$
264.288.10-264.j.1.37 $264$ $2$ $2$ $10$
264.288.10-264.j.2.5 $264$ $2$ $2$ $10$
264.288.10-264.j.2.37 $264$ $2$ $2$ $10$
264.432.16-132.c.1.83 $264$ $3$ $3$ $16$