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Properties

Label 16.48.1.bg.1
Level $16$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $16$ $\SL_2$-level: $8$ Newform level: $256$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $8^{6}$ Cusp orbits $2\cdot4$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 8H1
Sutherland and Zywina (SZ) label: 8H1-16k
Rouse and Zureick-Brown (RZB) label: X302
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.48.1.139

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&11\\10&5\end{bmatrix}$, $\begin{bmatrix}11&10\\12&15\end{bmatrix}$, $\begin{bmatrix}13&9\\12&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 16-isogeny field degree: $8$
Cyclic 16-torsion field degree: $64$
Full 16-torsion field degree: $512$

Jacobian

Conductor: $2^{8}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 256.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 13x - 21 $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

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 -2^6\,\frac{1728x^{2}y^{30}+451584x^{2}y^{29}z+68809984x^{2}y^{28}z^{2}+7351812096x^{2}y^{27}z^{3}-47462907904x^{2}y^{26}z^{4}-20054470033408x^{2}y^{25}z^{5}-80894428381184x^{2}y^{24}z^{6}+25930306703851520x^{2}y^{23}z^{7}+515521405836591104x^{2}y^{22}z^{8}-53936041805106642944x^{2}y^{21}z^{9}+425531262006967926784x^{2}y^{20}z^{10}+50654187719664825532416x^{2}y^{19}z^{11}-1842027646025860808441856x^{2}y^{18}z^{12}+12944001752502201256771584x^{2}y^{17}z^{13}+1023538887605954227382255616x^{2}y^{16}z^{14}-47553841412012806385027776512x^{2}y^{15}z^{15}+1221564375240559269564704096256x^{2}y^{14}z^{16}-22908765323416958544503031988224x^{2}y^{13}z^{17}+341479113919498835194876937633792x^{2}y^{12}z^{18}-4209857580281496347066405757124608x^{2}y^{11}z^{19}+43873060840864893896246016220332032x^{2}y^{10}z^{20}-391360075208112371305825277623402496x^{2}y^{9}z^{21}+3007486589430287466929522401372798976x^{2}y^{8}z^{22}-19944864142962481126320285970331598848x^{2}y^{7}z^{23}+113839388270475055112104449062792593408x^{2}y^{6}z^{24}-554933355679969066413879242327555309568x^{2}y^{5}z^{25}+2277241034673146664920366650477961543680x^{2}y^{4}z^{26}-7674682889519107800512083047009300250624x^{2}y^{3}z^{27}+20323244716878305192306000190178760589312x^{2}y^{2}z^{28}-38777617978829155744424504706303472435200x^{2}yz^{29}+42553386927732778367677865725802236084224x^{2}z^{30}-21888xy^{30}z-9421824xy^{29}z^{2}+35237376xy^{28}z^{3}-9802055680xy^{27}z^{4}+6265345433600xy^{26}z^{5}-84090893107200xy^{25}z^{6}-9716575491457024xy^{24}z^{7}+101822924956106752xy^{23}z^{8}+15189798819618357248xy^{22}z^{9}-399770257078993551360xy^{21}z^{10}-9288564555746094612480xy^{20}z^{11}+672637333842607118221312xy^{19}z^{12}-11861628851679052376834048xy^{18}z^{13}-167301646003620952986877952xy^{17}z^{14}+15236279678942645467924660224xy^{16}z^{15}-482832303337682780597910503424xy^{15}z^{16}+10416961592264008649856597884928xy^{14}z^{17}-174316043601071505443661381894144xy^{13}z^{18}+2386899862122562306748347098071040xy^{12}z^{19}-27507014218765705422006079933382656xy^{11}z^{20}+271158134084172182484074638480506880xy^{10}z^{21}-2307943867733334528681052495304196096xy^{9}z^{22}+17037355363951125346270345837047447552xy^{8}z^{23}-109129255842258855524732893132499189760xy^{7}z^{24}+604351803316337528900369249059907567616xy^{6}z^{25}-2869739870952017005353149525742608973824xy^{5}z^{26}+11512409420062885854560712560382073372672xy^{4}z^{27}-38060535492789135309528638527913571909632xy^{3}z^{28}+99230942928228475425784871936173145063424xy^{2}z^{29}-187234902482024224747655466922915607347200xyz^{30}+205465927691684891324560364085374512791552xz^{31}+27y^{32}+10368y^{31}z-611648y^{30}z^{2}-192172032y^{29}z^{3}-24126358272y^{28}z^{4}+91925495808y^{27}z^{5}+6485353340928y^{26}z^{6}+3174630237929472y^{25}z^{7}-92484164526997504y^{24}z^{8}-3116099295548276736y^{23}z^{9}+153037285773543473152y^{22}z^{10}+1099923138430252351488y^{21}z^{11}-196276539282546348785664y^{20}z^{12}+4654912892161910726721536y^{19}z^{13}+16387557858532952530485248y^{18}z^{14}-4153591209759274963847610368y^{17}z^{15}+147063540890012050392390565888y^{16}z^{16}-3336020407613930532064039272448y^{15}z^{17}+57509942369898232070556191031296y^{14}z^{18}-803422144313542592725684613808128y^{13}z^{19}+9398819554241966730687245367377920y^{12}z^{20}-93845969703321690736095255599251456y^{11}z^{21}+809048461759868061443168667730706432y^{10}z^{22}-6061833931979634252965223750986366976y^{9}z^{23}+39584178022580746492420758233591840768y^{8}z^{24}-225152764906138705609678529481232351232y^{7}z^{25}+1110998599771352551679528362586138476544y^{6}z^{26}-4717220695591417533609816480411256094720y^{5}z^{27}+16994702853307053879291680336089070960640y^{4}z^{28}-50788311146564274534466534603130482458624y^{3}z^{29}+121015425567119651779285072843110032080896y^{2}z^{30}-212706145636610272543145858412015570124800yz^{31}+233417300725459668792084195761383834386432z^{32}}{128x^{2}y^{30}-2756864x^{2}y^{28}z^{2}+463634432x^{2}y^{26}z^{4}+795511685120x^{2}y^{24}z^{6}-630809446842368x^{2}y^{22}z^{8}+277124551068876800x^{2}y^{20}z^{10}-83601154186718216192x^{2}y^{18}z^{12}+16043074226840154406912x^{2}y^{16}z^{14}-1306433362507541166161920x^{2}y^{14}z^{16}-165315989922644533593505792x^{2}y^{12}z^{18}+53480740841842931439711027200x^{2}y^{10}z^{20}-4458754864003565948946209570816x^{2}y^{8}z^{22}-69177709352601989045667487547392x^{2}y^{6}z^{24}+27362516897174091319582902666133504x^{2}y^{4}z^{26}-500049320183361667189655334452985856x^{2}y^{2}z^{28}-50433643766201811970135666768918609920x^{2}z^{30}-6720xy^{30}z+15391232xy^{28}z^{3}-15368069120xy^{26}z^{5}+7986830245888xy^{24}z^{7}-3241671881916416xy^{22}z^{9}+1330160827579236352xy^{20}z^{11}-479470637452728729600xy^{18}z^{13}+119079170607821354958848xy^{16}z^{15}-16419670523081775487385600xy^{14}z^{17}+428384582896670949588860928xy^{12}z^{19}+215733155788503260379679293440xy^{10}z^{21}-27958283220715157304784594141184xy^{8}z^{23}+322489386170371090248971671568384xy^{6}z^{25}+130097725520745774622367547693989888xy^{4}z^{27}-3720095404275128261128107704102944768xy^{2}z^{29}-243515173560515425612221816728211947520xz^{31}-y^{32}+186048y^{30}z^{2}+35944192y^{28}z^{4}-11210448896y^{26}z^{6}-30167290281984y^{24}z^{8}+20596318602788864y^{22}z^{10}-5756425907745062912y^{20}z^{12}+390228011579179270144y^{18}z^{14}+241424293224642200469504y^{16}z^{16}-82157956730206166724902912y^{14}z^{18}+10716631011403363059253641216y^{12}z^{20}-315114065862838130301974085632y^{10}z^{22}-66830233301158821989754888781824y^{8}z^{24}+5534380180712195744228098682388480y^{6}z^{26}+98801627533044266485341608638152704y^{4}z^{28}-14045678460762041665285516609313570816y^{2}z^{30}-276642726785729973827810932134013566976z^{32}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.bc.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.24.0.k.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.24.1.f.1 $16$ $2$ $2$ $1$ $1$ 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
16.96.3.b.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.bg.1 $16$ $2$ $2$ $3$ $2$ $1^{2}$
16.96.3.cm.2 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.cx.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.rt.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.rx.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.sn.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.sr.1 $48$ $2$ $2$ $3$ $3$ $1^{2}$
48.144.7.pa.1 $48$ $3$ $3$ $7$ $6$ $1^{2}\cdot2^{2}$
48.192.11.kc.1 $48$ $4$ $4$ $11$ $3$ $1^{8}\cdot2$
80.96.3.vd.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.vh.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.vx.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.wb.1 $80$ $2$ $2$ $3$ $?$ not computed
80.240.17.hu.1 $80$ $5$ $5$ $17$ $?$ not computed
80.288.17.bau.1 $80$ $6$ $6$ $17$ $?$ not computed
112.96.3.qt.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.qx.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.rn.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.rr.1 $112$ $2$ $2$ $3$ $?$ not computed
176.96.3.qt.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.qx.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.rn.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.rr.1 $176$ $2$ $2$ $3$ $?$ not computed
208.96.3.vd.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.vh.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.vx.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.wb.1 $208$ $2$ $2$ $3$ $?$ not computed
240.96.3.fdf.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fdn.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fet.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ffb.1 $240$ $2$ $2$ $3$ $?$ not computed
272.96.3.vd.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.vh.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.vx.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.wb.1 $272$ $2$ $2$ $3$ $?$ not computed
304.96.3.qt.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.qx.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.rn.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.rr.1 $304$ $2$ $2$ $3$ $?$ not computed