Normalized defining polynomial
\( x^{20} - x^{19} + 55 x^{18} - 278 x^{17} + 2263 x^{16} - 11313 x^{15} + 79085 x^{14} + \cdots + 197469154012 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(393644681588957236910214274346592274945338834944\)
\(\medspace = 2^{16}\cdot 7^{15}\cdot 11^{15}\cdot 13^{13}\)
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| Root discriminant: | \(239.75\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{1001}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-154 +10 \sqrt{77}})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{208}a^{15}+\frac{9}{208}a^{14}+\frac{1}{208}a^{13}-\frac{27}{104}a^{12}+\frac{9}{208}a^{11}-\frac{51}{208}a^{10}-\frac{23}{208}a^{9}+\frac{3}{104}a^{8}+\frac{15}{208}a^{7}-\frac{71}{208}a^{6}+\frac{79}{208}a^{5}-\frac{41}{104}a^{4}+\frac{3}{8}a^{3}-\frac{1}{8}a-\frac{1}{8}$, $\frac{1}{416}a^{16}-\frac{1}{416}a^{15}-\frac{89}{416}a^{14}-\frac{2}{13}a^{13}+\frac{133}{416}a^{12}-\frac{141}{416}a^{11}+\frac{71}{416}a^{10}+\frac{7}{104}a^{9}+\frac{163}{416}a^{8}+\frac{15}{32}a^{7}+\frac{165}{416}a^{6}+\frac{21}{52}a^{5}-\frac{71}{208}a^{4}+\frac{1}{8}a^{3}+\frac{7}{16}a^{2}+\frac{1}{16}a-\frac{3}{8}$, $\frac{1}{416}a^{17}-\frac{175}{416}a^{14}+\frac{159}{416}a^{13}+\frac{31}{104}a^{12}-\frac{23}{104}a^{11}+\frac{85}{416}a^{10}+\frac{201}{416}a^{9}+\frac{33}{208}a^{8}+\frac{23}{208}a^{7}+\frac{183}{416}a^{6}+\frac{2}{13}a^{5}+\frac{9}{208}a^{4}+\frac{7}{16}a^{3}-\frac{1}{2}a^{2}+\frac{1}{16}a$, $\frac{1}{34750796704}a^{18}+\frac{759922}{1085962397}a^{17}-\frac{2056787}{2171924794}a^{16}+\frac{836495}{2044164512}a^{15}+\frac{373168625}{2673138208}a^{14}+\frac{6662843853}{17375398352}a^{13}-\frac{36139306}{1085962397}a^{12}-\frac{298327645}{2044164512}a^{11}-\frac{4377766529}{34750796704}a^{10}-\frac{688822631}{2171924794}a^{9}+\frac{4845645}{78621712}a^{8}-\frac{39574985}{181941344}a^{7}-\frac{3613855905}{17375398352}a^{6}-\frac{1441715835}{8687699176}a^{5}+\frac{7534776109}{17375398352}a^{4}+\frac{300463145}{668284552}a^{3}+\frac{140451137}{1336569104}a^{2}-\frac{330082031}{668284552}a+\frac{263495681}{668284552}$, $\frac{1}{47\cdots 56}a^{19}-\frac{40\cdots 15}{47\cdots 56}a^{18}+\frac{88\cdots 95}{47\cdots 56}a^{17}-\frac{24\cdots 71}{47\cdots 56}a^{16}+\frac{35\cdots 85}{11\cdots 64}a^{15}-\frac{78\cdots 79}{23\cdots 28}a^{14}+\frac{38\cdots 81}{27\cdots 68}a^{13}+\frac{18\cdots 33}{47\cdots 56}a^{12}+\frac{36\cdots 07}{23\cdots 28}a^{11}+\frac{91\cdots 99}{23\cdots 28}a^{10}+\frac{23\cdots 67}{47\cdots 56}a^{9}-\frac{99\cdots 89}{47\cdots 56}a^{8}-\frac{12\cdots 07}{47\cdots 56}a^{7}+\frac{46\cdots 47}{47\cdots 56}a^{6}+\frac{68\cdots 35}{29\cdots 66}a^{5}+\frac{30\cdots 65}{11\cdots 64}a^{4}-\frac{42\cdots 01}{90\cdots 28}a^{3}+\frac{12\cdots 89}{10\cdots 68}a^{2}-\frac{42\cdots 61}{18\cdots 56}a+\frac{20\cdots 84}{11\cdots 41}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}\times C_{8}$, which has order $64$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{4}\times C_{8}$, which has order $64$ (assuming GRH) |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{63\cdots 05}{14\cdots 33}a^{19}+\frac{17\cdots 89}{29\cdots 66}a^{18}+\frac{36\cdots 78}{14\cdots 33}a^{17}-\frac{18\cdots 51}{29\cdots 66}a^{16}+\frac{11\cdots 64}{14\cdots 33}a^{15}-\frac{41\cdots 54}{14\cdots 33}a^{14}+\frac{22\cdots 49}{86\cdots 49}a^{13}-\frac{29\cdots 17}{29\cdots 66}a^{12}+\frac{74\cdots 74}{14\cdots 33}a^{11}-\frac{33\cdots 27}{14\cdots 33}a^{10}+\frac{14\cdots 80}{14\cdots 33}a^{9}-\frac{97\cdots 17}{29\cdots 66}a^{8}+\frac{16\cdots 88}{14\cdots 33}a^{7}-\frac{11\cdots 73}{29\cdots 66}a^{6}+\frac{17\cdots 22}{14\cdots 33}a^{5}-\frac{37\cdots 74}{14\cdots 33}a^{4}+\frac{68\cdots 50}{11\cdots 41}a^{3}-\frac{10\cdots 52}{66\cdots 73}a^{2}+\frac{33\cdots 62}{11\cdots 41}a-\frac{28\cdots 13}{11\cdots 41}$, $\frac{22\cdots 71}{29\cdots 66}a^{19}-\frac{53\cdots 49}{29\cdots 66}a^{18}-\frac{66\cdots 63}{14\cdots 33}a^{17}+\frac{84\cdots 26}{14\cdots 33}a^{16}-\frac{39\cdots 07}{29\cdots 66}a^{15}+\frac{50\cdots 09}{14\cdots 33}a^{14}-\frac{61\cdots 53}{14\cdots 33}a^{13}+\frac{19\cdots 80}{14\cdots 33}a^{12}-\frac{21\cdots 21}{29\cdots 66}a^{11}+\frac{45\cdots 05}{14\cdots 33}a^{10}-\frac{20\cdots 04}{14\cdots 33}a^{9}+\frac{61\cdots 76}{14\cdots 33}a^{8}-\frac{21\cdots 06}{14\cdots 33}a^{7}+\frac{14\cdots 71}{29\cdots 66}a^{6}-\frac{19\cdots 14}{14\cdots 33}a^{5}+\frac{42\cdots 78}{14\cdots 33}a^{4}-\frac{74\cdots 38}{11\cdots 41}a^{3}+\frac{19\cdots 86}{11\cdots 41}a^{2}-\frac{36\cdots 80}{11\cdots 41}a+\frac{44\cdots 87}{11\cdots 41}$, $\frac{29\cdots 69}{38\cdots 72}a^{19}+\frac{25\cdots 49}{77\cdots 44}a^{18}-\frac{19\cdots 77}{19\cdots 36}a^{17}+\frac{13\cdots 99}{77\cdots 44}a^{16}-\frac{23\cdots 45}{19\cdots 36}a^{15}+\frac{16\cdots 91}{19\cdots 36}a^{14}-\frac{18\cdots 11}{38\cdots 72}a^{13}+\frac{23\cdots 63}{77\cdots 44}a^{12}-\frac{78\cdots 19}{48\cdots 34}a^{11}+\frac{29\cdots 85}{38\cdots 72}a^{10}-\frac{32\cdots 73}{96\cdots 68}a^{9}+\frac{11\cdots 23}{77\cdots 44}a^{8}-\frac{14\cdots 19}{24\cdots 17}a^{7}+\frac{14\cdots 87}{77\cdots 44}a^{6}-\frac{12\cdots 99}{19\cdots 36}a^{5}+\frac{39\cdots 19}{19\cdots 36}a^{4}-\frac{82\cdots 39}{14\cdots 72}a^{3}+\frac{20\cdots 44}{18\cdots 09}a^{2}-\frac{42\cdots 01}{29\cdots 44}a+\frac{18\cdots 31}{18\cdots 09}$, $\frac{34\cdots 23}{11\cdots 64}a^{19}+\frac{26\cdots 01}{47\cdots 56}a^{18}-\frac{64\cdots 83}{47\cdots 56}a^{17}+\frac{47\cdots 05}{47\cdots 56}a^{16}-\frac{13\cdots 17}{23\cdots 28}a^{15}+\frac{17\cdots 13}{47\cdots 56}a^{14}-\frac{10\cdots 51}{47\cdots 56}a^{13}+\frac{58\cdots 01}{47\cdots 56}a^{12}-\frac{11\cdots 63}{23\cdots 28}a^{11}+\frac{11\cdots 27}{47\cdots 56}a^{10}-\frac{49\cdots 79}{47\cdots 56}a^{9}+\frac{18\cdots 75}{47\cdots 56}a^{8}-\frac{30\cdots 77}{23\cdots 28}a^{7}+\frac{10\cdots 69}{23\cdots 28}a^{6}-\frac{16\cdots 69}{11\cdots 64}a^{5}+\frac{73\cdots 39}{23\cdots 28}a^{4}-\frac{13\cdots 37}{18\cdots 56}a^{3}+\frac{75\cdots 99}{45\cdots 64}a^{2}-\frac{80\cdots 91}{26\cdots 92}a+\frac{18\cdots 95}{45\cdots 64}$, $\frac{54\cdots 93}{47\cdots 56}a^{19}+\frac{18\cdots 65}{27\cdots 68}a^{18}-\frac{34\cdots 89}{58\cdots 32}a^{17}+\frac{14\cdots 37}{47\cdots 56}a^{16}-\frac{26\cdots 39}{11\cdots 64}a^{15}+\frac{53\cdots 21}{47\cdots 56}a^{14}-\frac{92\cdots 51}{11\cdots 64}a^{13}+\frac{18\cdots 93}{47\cdots 56}a^{12}-\frac{39\cdots 37}{23\cdots 28}a^{11}+\frac{37\cdots 57}{47\cdots 56}a^{10}-\frac{20\cdots 63}{58\cdots 32}a^{9}+\frac{57\cdots 81}{47\cdots 56}a^{8}-\frac{11\cdots 59}{27\cdots 68}a^{7}+\frac{16\cdots 43}{11\cdots 64}a^{6}-\frac{25\cdots 39}{58\cdots 32}a^{5}+\frac{22\cdots 29}{23\cdots 28}a^{4}-\frac{37\cdots 61}{18\cdots 56}a^{3}+\frac{98\cdots 47}{18\cdots 56}a^{2}-\frac{90\cdots 33}{90\cdots 28}a+\frac{80\cdots 92}{11\cdots 41}$, $\frac{50\cdots 05}{47\cdots 56}a^{19}-\frac{15\cdots 51}{47\cdots 56}a^{18}+\frac{17\cdots 21}{47\cdots 56}a^{17}-\frac{70\cdots 18}{14\cdots 33}a^{16}+\frac{81\cdots 55}{47\cdots 56}a^{15}-\frac{69\cdots 25}{47\cdots 56}a^{14}+\frac{36\cdots 01}{47\cdots 56}a^{13}-\frac{53\cdots 93}{11\cdots 64}a^{12}+\frac{81\cdots 09}{47\cdots 56}a^{11}-\frac{41\cdots 29}{47\cdots 56}a^{10}+\frac{19\cdots 19}{47\cdots 56}a^{9}-\frac{19\cdots 90}{14\cdots 33}a^{8}+\frac{52\cdots 13}{11\cdots 64}a^{7}-\frac{39\cdots 89}{23\cdots 28}a^{6}+\frac{10\cdots 93}{23\cdots 28}a^{5}-\frac{24\cdots 47}{23\cdots 28}a^{4}+\frac{11\cdots 85}{45\cdots 64}a^{3}-\frac{54\cdots 83}{90\cdots 28}a^{2}+\frac{99\cdots 23}{90\cdots 28}a-\frac{26\cdots 23}{22\cdots 82}$, $\frac{15\cdots 79}{11\cdots 64}a^{19}+\frac{58\cdots 89}{11\cdots 64}a^{18}-\frac{23\cdots 69}{47\cdots 56}a^{17}+\frac{10\cdots 65}{47\cdots 56}a^{16}-\frac{67\cdots 55}{47\cdots 56}a^{15}+\frac{19\cdots 17}{58\cdots 32}a^{14}-\frac{12\cdots 61}{47\cdots 56}a^{13}+\frac{41\cdots 33}{47\cdots 56}a^{12}+\frac{33\cdots 21}{47\cdots 56}a^{11}-\frac{19\cdots 25}{11\cdots 64}a^{10}+\frac{39\cdots 45}{47\cdots 56}a^{9}-\frac{26\cdots 83}{47\cdots 56}a^{8}+\frac{15\cdots 59}{47\cdots 56}a^{7}-\frac{14\cdots 67}{11\cdots 64}a^{6}+\frac{81\cdots 99}{23\cdots 28}a^{5}-\frac{11\cdots 93}{11\cdots 64}a^{4}+\frac{48\cdots 41}{18\cdots 56}a^{3}-\frac{11\cdots 05}{18\cdots 56}a^{2}+\frac{88\cdots 13}{90\cdots 28}a-\frac{18\cdots 95}{22\cdots 82}$, $\frac{56\cdots 75}{47\cdots 56}a^{19}+\frac{26\cdots 13}{47\cdots 56}a^{18}-\frac{20\cdots 51}{47\cdots 56}a^{17}+\frac{47\cdots 39}{11\cdots 64}a^{16}-\frac{57\cdots 13}{47\cdots 56}a^{15}+\frac{56\cdots 55}{47\cdots 56}a^{14}-\frac{25\cdots 35}{36\cdots 12}a^{13}+\frac{17\cdots 85}{58\cdots 32}a^{12}-\frac{65\cdots 67}{47\cdots 56}a^{11}+\frac{33\cdots 63}{47\cdots 56}a^{10}-\frac{13\cdots 45}{47\cdots 56}a^{9}+\frac{10\cdots 33}{11\cdots 64}a^{8}-\frac{19\cdots 79}{58\cdots 32}a^{7}+\frac{26\cdots 61}{23\cdots 28}a^{6}-\frac{41\cdots 03}{13\cdots 12}a^{5}+\frac{16\cdots 77}{23\cdots 28}a^{4}-\frac{75\cdots 41}{45\cdots 64}a^{3}+\frac{26\cdots 75}{69\cdots 56}a^{2}-\frac{57\cdots 43}{90\cdots 28}a+\frac{13\cdots 49}{22\cdots 82}$, $\frac{12\cdots 77}{47\cdots 56}a^{19}-\frac{11\cdots 49}{27\cdots 68}a^{18}+\frac{24\cdots 87}{23\cdots 28}a^{17}-\frac{58\cdots 47}{47\cdots 56}a^{16}+\frac{94\cdots 87}{11\cdots 64}a^{15}-\frac{34\cdots 01}{47\cdots 56}a^{14}+\frac{20\cdots 31}{58\cdots 32}a^{13}-\frac{81\cdots 95}{47\cdots 56}a^{12}+\frac{22\cdots 55}{23\cdots 28}a^{11}-\frac{21\cdots 25}{47\cdots 56}a^{10}+\frac{20\cdots 29}{11\cdots 64}a^{9}-\frac{28\cdots 23}{47\cdots 56}a^{8}+\frac{64\cdots 33}{27\cdots 68}a^{7}-\frac{88\cdots 11}{11\cdots 64}a^{6}+\frac{45\cdots 01}{23\cdots 28}a^{5}-\frac{10\cdots 73}{23\cdots 28}a^{4}+\frac{15\cdots 79}{13\cdots 12}a^{3}-\frac{44\cdots 23}{18\cdots 56}a^{2}+\frac{33\cdots 11}{90\cdots 28}a-\frac{28\cdots 13}{90\cdots 28}$
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| Regulator: | \( 152693517470566.22 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 152693517470566.22 \cdot 64}{2\cdot\sqrt{393644681588957236910214274346592274945338834944}}\cr\approx \mathstrut & 0.746821938963927 \end{aligned}\] (assuming GRH)
Galois group
$C_{20}:C_4$ (as 20T18):
| A solvable group of order 80 |
| The 14 conjugacy class representatives for $C_{20}:C_4$ |
| Character table for $C_{20}:C_4$ |
Intermediate fields
| \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-154 +10 \sqrt{77}})\), 5.1.35152.1, deg 10 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | R | R | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | $20$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
|
\(7\)
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.16.12.1 | $x^{16} + 20 x^{14} + 16 x^{13} + 162 x^{12} + 240 x^{11} + 776 x^{10} + 1344 x^{9} + 2539 x^{8} + 3696 x^{7} + 5016 x^{6} + 5312 x^{5} + 4594 x^{4} + 2928 x^{3} + 1404 x^{2} + 432 x + 88$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(11\)
| 11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 11.16.12.1 | $x^{16} + 32 x^{14} + 40 x^{13} + 392 x^{12} + 960 x^{11} + 2840 x^{10} + 7920 x^{9} + 15256 x^{8} + 28320 x^{7} + 45280 x^{6} + 47840 x^{5} + 30768 x^{4} + 11840 x^{3} + 2656 x^{2} + 320 x + 27$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |