Normalized defining polynomial
\( x^{20} - 7 x^{18} - 16 x^{17} + 18 x^{16} + 100 x^{15} + 102 x^{14} - 222 x^{13} - 93 x^{12} + \cdots - 423 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[2, 9]$ |
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| Discriminant: |
\(-2115524933097244001040375742464\)
\(\medspace = -\,2^{20}\cdot 3^{18}\cdot 13^{5}\cdot 107^{5}\)
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| |
| Root discriminant: | \(32.83\) |
| |
| Galois root discriminant: | $2\cdot 3^{43/36}13^{5/6}107^{5/6}\approx 3092.8873527512997$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(107\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1391}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{9}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{10}+\frac{1}{3}a^{4}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{15}+\frac{1}{9}a^{14}-\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{3}a^{10}-\frac{2}{9}a^{9}-\frac{4}{9}a^{8}+\frac{1}{3}a^{7}-\frac{2}{9}a^{6}-\frac{1}{3}a^{5}$, $\frac{1}{2691}a^{18}+\frac{128}{2691}a^{17}+\frac{35}{2691}a^{16}-\frac{292}{2691}a^{15}-\frac{265}{2691}a^{14}+\frac{389}{2691}a^{13}+\frac{58}{897}a^{12}-\frac{374}{2691}a^{11}-\frac{80}{207}a^{10}-\frac{44}{2691}a^{9}+\frac{883}{2691}a^{8}+\frac{1234}{2691}a^{7}-\frac{571}{2691}a^{6}+\frac{9}{23}a^{5}-\frac{103}{299}a^{4}+\frac{61}{897}a^{3}+\frac{27}{299}a^{2}+\frac{96}{299}a+\frac{44}{299}$, $\frac{1}{59\cdots 33}a^{19}-\frac{96\cdots 10}{59\cdots 33}a^{18}+\frac{18\cdots 40}{59\cdots 33}a^{17}+\frac{67\cdots 24}{59\cdots 33}a^{16}+\frac{22\cdots 26}{19\cdots 11}a^{15}-\frac{73\cdots 31}{25\cdots 71}a^{14}-\frac{16\cdots 50}{19\cdots 11}a^{13}+\frac{31\cdots 10}{59\cdots 33}a^{12}-\frac{49\cdots 82}{59\cdots 33}a^{11}+\frac{17\cdots 35}{59\cdots 33}a^{10}+\frac{31\cdots 11}{19\cdots 11}a^{9}+\frac{87\cdots 22}{59\cdots 33}a^{8}+\frac{12\cdots 84}{59\cdots 33}a^{7}-\frac{19\cdots 45}{59\cdots 33}a^{6}-\frac{29\cdots 51}{65\cdots 37}a^{5}-\frac{14\cdots 67}{65\cdots 37}a^{4}+\frac{63\cdots 52}{50\cdots 49}a^{3}+\frac{23\cdots 53}{65\cdots 37}a^{2}+\frac{22\cdots 38}{65\cdots 37}a+\frac{12\cdots 29}{65\cdots 37}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
|
Unit group
| Rank: | $10$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{10\cdots 91}{59\cdots 33}a^{19}-\frac{17\cdots 51}{19\cdots 11}a^{18}-\frac{69\cdots 66}{59\cdots 33}a^{17}-\frac{14\cdots 46}{59\cdots 33}a^{16}+\frac{25\cdots 22}{59\cdots 33}a^{15}+\frac{95\cdots 62}{59\cdots 33}a^{14}+\frac{78\cdots 30}{59\cdots 33}a^{13}-\frac{27\cdots 73}{59\cdots 33}a^{12}+\frac{45\cdots 87}{59\cdots 33}a^{11}+\frac{11\cdots 45}{59\cdots 33}a^{10}-\frac{34\cdots 32}{59\cdots 33}a^{9}-\frac{58\cdots 86}{59\cdots 33}a^{8}+\frac{31\cdots 45}{65\cdots 37}a^{7}-\frac{39\cdots 77}{59\cdots 33}a^{6}+\frac{31\cdots 60}{85\cdots 57}a^{5}-\frac{58\cdots 17}{19\cdots 11}a^{4}-\frac{37\cdots 52}{19\cdots 11}a^{3}+\frac{71\cdots 95}{65\cdots 37}a^{2}-\frac{25\cdots 05}{65\cdots 37}a-\frac{29\cdots 11}{65\cdots 37}$, $\frac{21\cdots 39}{19\cdots 11}a^{19}-\frac{49\cdots 18}{59\cdots 33}a^{18}-\frac{47\cdots 64}{59\cdots 33}a^{17}-\frac{68\cdots 25}{59\cdots 33}a^{16}+\frac{75\cdots 67}{19\cdots 11}a^{15}+\frac{61\cdots 00}{59\cdots 33}a^{14}+\frac{86\cdots 21}{59\cdots 33}a^{13}-\frac{24\cdots 57}{59\cdots 33}a^{12}-\frac{13\cdots 09}{59\cdots 33}a^{11}+\frac{88\cdots 57}{59\cdots 33}a^{10}-\frac{47\cdots 21}{65\cdots 37}a^{9}-\frac{10\cdots 42}{59\cdots 33}a^{8}+\frac{68\cdots 99}{59\cdots 33}a^{7}+\frac{13\cdots 76}{59\cdots 33}a^{6}+\frac{68\cdots 92}{28\cdots 19}a^{5}-\frac{13\cdots 89}{65\cdots 37}a^{4}-\frac{60\cdots 84}{19\cdots 11}a^{3}+\frac{25\cdots 86}{65\cdots 37}a^{2}-\frac{13\cdots 07}{65\cdots 37}a-\frac{89\cdots 76}{65\cdots 37}$, $\frac{12\cdots 53}{65\cdots 37}a^{19}+\frac{12\cdots 87}{59\cdots 33}a^{18}-\frac{14\cdots 36}{59\cdots 33}a^{17}-\frac{68\cdots 27}{59\cdots 33}a^{16}-\frac{39\cdots 25}{19\cdots 11}a^{15}+\frac{37\cdots 00}{59\cdots 33}a^{14}+\frac{80\cdots 86}{59\cdots 33}a^{13}+\frac{31\cdots 40}{59\cdots 33}a^{12}-\frac{33\cdots 86}{59\cdots 33}a^{11}+\frac{26\cdots 18}{59\cdots 33}a^{10}+\frac{12\cdots 18}{65\cdots 37}a^{9}-\frac{10\cdots 28}{59\cdots 33}a^{8}+\frac{90\cdots 70}{59\cdots 33}a^{7}+\frac{22\cdots 22}{59\cdots 33}a^{6}+\frac{46\cdots 27}{85\cdots 57}a^{5}+\frac{28\cdots 27}{65\cdots 37}a^{4}-\frac{11\cdots 66}{19\cdots 11}a^{3}+\frac{16\cdots 27}{65\cdots 37}a^{2}-\frac{28\cdots 23}{65\cdots 37}a-\frac{29\cdots 41}{65\cdots 37}$, $\frac{12\cdots 05}{59\cdots 33}a^{19}-\frac{75\cdots 31}{19\cdots 11}a^{18}-\frac{75\cdots 51}{59\cdots 33}a^{17}-\frac{19\cdots 57}{59\cdots 33}a^{16}+\frac{67\cdots 95}{19\cdots 11}a^{15}+\frac{10\cdots 12}{59\cdots 33}a^{14}+\frac{13\cdots 47}{59\cdots 33}a^{13}-\frac{26\cdots 78}{65\cdots 37}a^{12}-\frac{30\cdots 81}{59\cdots 33}a^{11}+\frac{11\cdots 92}{59\cdots 33}a^{10}+\frac{14\cdots 52}{19\cdots 11}a^{9}-\frac{18\cdots 45}{45\cdots 41}a^{8}-\frac{42\cdots 13}{19\cdots 11}a^{7}-\frac{11\cdots 77}{19\cdots 11}a^{6}+\frac{32\cdots 46}{85\cdots 57}a^{5}-\frac{43\cdots 98}{19\cdots 11}a^{4}-\frac{32\cdots 51}{19\cdots 11}a^{3}-\frac{59\cdots 46}{65\cdots 37}a^{2}-\frac{15\cdots 26}{50\cdots 49}a-\frac{14\cdots 22}{65\cdots 37}$, $\frac{38\cdots 57}{65\cdots 37}a^{19}+\frac{15\cdots 16}{59\cdots 33}a^{18}-\frac{23\cdots 52}{59\cdots 33}a^{17}-\frac{72\cdots 49}{59\cdots 33}a^{16}+\frac{39\cdots 70}{59\cdots 33}a^{15}+\frac{40\cdots 63}{59\cdots 33}a^{14}+\frac{59\cdots 34}{59\cdots 33}a^{13}-\frac{24\cdots 15}{19\cdots 11}a^{12}-\frac{11\cdots 99}{59\cdots 33}a^{11}+\frac{33\cdots 37}{59\cdots 33}a^{10}+\frac{34\cdots 14}{59\cdots 33}a^{9}-\frac{25\cdots 05}{59\cdots 33}a^{8}-\frac{34\cdots 92}{59\cdots 33}a^{7}+\frac{25\cdots 19}{59\cdots 33}a^{6}+\frac{10\cdots 09}{85\cdots 57}a^{5}+\frac{31\cdots 26}{65\cdots 37}a^{4}-\frac{99\cdots 48}{65\cdots 37}a^{3}-\frac{51\cdots 20}{65\cdots 37}a^{2}+\frac{73\cdots 41}{65\cdots 37}a-\frac{16\cdots 12}{65\cdots 37}$, $\frac{33\cdots 52}{53\cdots 63}a^{19}-\frac{81\cdots 21}{47\cdots 67}a^{18}-\frac{13\cdots 07}{47\cdots 67}a^{17}-\frac{64\cdots 29}{47\cdots 67}a^{16}+\frac{50\cdots 01}{15\cdots 89}a^{15}+\frac{10\cdots 67}{47\cdots 67}a^{14}-\frac{22\cdots 53}{47\cdots 67}a^{13}-\frac{11\cdots 32}{47\cdots 67}a^{12}+\frac{11\cdots 08}{36\cdots 59}a^{11}+\frac{20\cdots 15}{47\cdots 67}a^{10}-\frac{67\cdots 52}{53\cdots 63}a^{9}+\frac{26\cdots 58}{47\cdots 67}a^{8}-\frac{29\cdots 31}{47\cdots 67}a^{7}+\frac{38\cdots 54}{36\cdots 59}a^{6}+\frac{24\cdots 43}{15\cdots 89}a^{5}-\frac{51\cdots 54}{15\cdots 89}a^{4}+\frac{43\cdots 37}{15\cdots 89}a^{3}-\frac{13\cdots 50}{53\cdots 63}a^{2}+\frac{17\cdots 85}{53\cdots 63}a+\frac{13\cdots 43}{40\cdots 51}$, $\frac{73\cdots 85}{19\cdots 11}a^{19}-\frac{14\cdots 53}{59\cdots 33}a^{18}-\frac{13\cdots 91}{59\cdots 33}a^{17}-\frac{28\cdots 36}{59\cdots 33}a^{16}+\frac{14\cdots 35}{15\cdots 47}a^{15}+\frac{18\cdots 55}{59\cdots 33}a^{14}+\frac{14\cdots 29}{59\cdots 33}a^{13}-\frac{56\cdots 36}{59\cdots 33}a^{12}+\frac{11\cdots 59}{59\cdots 33}a^{11}+\frac{22\cdots 41}{59\cdots 33}a^{10}-\frac{29\cdots 31}{19\cdots 11}a^{9}-\frac{62\cdots 11}{59\cdots 33}a^{8}+\frac{27\cdots 75}{59\cdots 33}a^{7}-\frac{26\cdots 90}{59\cdots 33}a^{6}+\frac{48\cdots 28}{65\cdots 37}a^{5}-\frac{29\cdots 40}{50\cdots 49}a^{4}-\frac{19\cdots 03}{19\cdots 11}a^{3}-\frac{15\cdots 35}{65\cdots 37}a^{2}-\frac{30\cdots 28}{65\cdots 37}a-\frac{51\cdots 25}{65\cdots 37}$, $\frac{46\cdots 98}{19\cdots 11}a^{19}+\frac{45\cdots 93}{19\cdots 11}a^{18}-\frac{10\cdots 37}{59\cdots 33}a^{17}-\frac{10\cdots 76}{19\cdots 11}a^{16}+\frac{87\cdots 53}{59\cdots 33}a^{15}+\frac{17\cdots 47}{59\cdots 33}a^{14}+\frac{88\cdots 57}{19\cdots 11}a^{13}-\frac{26\cdots 59}{59\cdots 33}a^{12}-\frac{56\cdots 46}{59\cdots 33}a^{11}+\frac{18\cdots 53}{65\cdots 37}a^{10}+\frac{18\cdots 29}{59\cdots 33}a^{9}-\frac{18\cdots 83}{59\cdots 33}a^{8}-\frac{16\cdots 01}{19\cdots 11}a^{7}+\frac{22\cdots 20}{59\cdots 33}a^{6}+\frac{37\cdots 63}{65\cdots 37}a^{5}+\frac{29\cdots 98}{65\cdots 37}a^{4}-\frac{18\cdots 51}{19\cdots 11}a^{3}-\frac{66\cdots 27}{28\cdots 19}a^{2}-\frac{10\cdots 03}{65\cdots 37}a-\frac{30\cdots 53}{65\cdots 37}$, $\frac{37\cdots 62}{59\cdots 33}a^{19}-\frac{70\cdots 49}{59\cdots 33}a^{18}-\frac{17\cdots 93}{59\cdots 33}a^{17}-\frac{27\cdots 58}{65\cdots 37}a^{16}+\frac{10\cdots 49}{45\cdots 41}a^{15}+\frac{16\cdots 99}{59\cdots 33}a^{14}-\frac{35\cdots 59}{59\cdots 33}a^{13}-\frac{34\cdots 18}{19\cdots 11}a^{12}+\frac{14\cdots 48}{59\cdots 33}a^{11}+\frac{90\cdots 17}{19\cdots 11}a^{10}-\frac{55\cdots 69}{59\cdots 33}a^{9}+\frac{15\cdots 40}{25\cdots 71}a^{8}-\frac{51\cdots 08}{59\cdots 33}a^{7}-\frac{55\cdots 91}{59\cdots 33}a^{6}+\frac{24\cdots 29}{19\cdots 11}a^{5}-\frac{36\cdots 42}{15\cdots 47}a^{4}+\frac{13\cdots 20}{65\cdots 37}a^{3}-\frac{92\cdots 86}{65\cdots 37}a^{2}+\frac{18\cdots 06}{65\cdots 37}a-\frac{10\cdots 54}{65\cdots 37}$, $\frac{78\cdots 16}{65\cdots 37}a^{19}-\frac{32\cdots 45}{19\cdots 11}a^{18}-\frac{56\cdots 12}{59\cdots 33}a^{17}-\frac{13\cdots 97}{19\cdots 11}a^{16}+\frac{34\cdots 13}{59\cdots 33}a^{15}+\frac{62\cdots 76}{59\cdots 33}a^{14}-\frac{58\cdots 12}{65\cdots 37}a^{13}-\frac{34\cdots 00}{59\cdots 33}a^{12}+\frac{13\cdots 53}{59\cdots 33}a^{11}+\frac{40\cdots 39}{19\cdots 11}a^{10}-\frac{10\cdots 90}{59\cdots 33}a^{9}-\frac{17\cdots 56}{59\cdots 33}a^{8}+\frac{53\cdots 06}{19\cdots 11}a^{7}+\frac{48\cdots 75}{59\cdots 33}a^{6}+\frac{43\cdots 07}{19\cdots 11}a^{5}-\frac{91\cdots 60}{19\cdots 11}a^{4}-\frac{31\cdots 83}{65\cdots 37}a^{3}+\frac{53\cdots 17}{65\cdots 37}a^{2}-\frac{99\cdots 22}{65\cdots 37}a+\frac{59\cdots 75}{65\cdots 37}$
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| Regulator: | \( 62306146.5873 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 62306146.5873 \cdot 1}{2\cdot\sqrt{2115524933097244001040375742464}}\cr\approx \mathstrut & 1.30758710577 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.\SOPlus(4,4)$ (as 20T1023):
| A non-solvable group of order 7372800 |
| The 324 conjugacy class representatives for $C_2^{10}.\SOPlus(4,4)$ |
| Character table for $C_2^{10}.\SOPlus(4,4)$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 10.6.38998285028352.1 |
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 | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | R | $20$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.2.10a1.2 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ |
| 2.5.2.10a1.2 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ | |
|
\(3\)
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
| 3.1.6.6a1.2 | $x^{6} + 3 x + 6$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $$[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$$ | |
| 3.1.6.6a1.2 | $x^{6} + 3 x + 6$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $$[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.1.3.2a1.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 13.1.3.2a1.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 13.6.1.0a1.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(107\)
| 107.2.1.0a1.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 107.2.1.0a1.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 107.2.1.0a1.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 107.1.2.1a1.2 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.0a1.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 107.1.3.2a1.1 | $x^{3} + 107$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 107.1.3.2a1.1 | $x^{3} + 107$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 107.4.1.0a1.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |