Number field 20.20.258866544008047715925279948537661096282306445312.1

• Label: 20.20.258...312.1
• Degree: $20$
• Signature: $(20, 0)$
• Discriminant: $2.589\times 10^{47}$
• Root discriminant: \(234.78\)
• Ramified primes: $2,17,53$
• Class number: $4$ (GRH)
• Class group: [4] (GRH)
• Galois group: $C_{20}:C_4$ (as 20T18)

Normalized defining polynomial

x^20 - 4*x^19 - 211*x^18 + 1138*x^17 + 12703*x^16 - 81202*x^15 - 279760*x^14 + 2366808*x^13 + 1504150*x^12 - 31913924*x^11 + 22954298*x^10 + 194714572*x^9 - 295344111*x^8 - 469380944*x^7 + 1107126565*x^6 + 182743098*x^5 - 1481772709*x^4 + 491417950*x^3 + 577682706*x^2 - 321816380*x + 25224604

Invariants

Degree:		$20$
Signature:		$(20, 0)$
Discriminant:		\(258866544008047715925279948537661096282306445312\) \(\medspace = 2^{16}\cdot 17^{15}\cdot 53^{14}\)
Root discriminant:		\(234.78\)
Galois root discriminant:		$2\cdot 17^{3/4}53^{3/4}\approx 328.90735776316194$
Ramified primes:		\(2\), \(17\), \(53\)
Discriminant root field:		\(\Q(\sqrt{17}) \)
$\Aut(K/\Q)$:		$C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{5}$, $\frac{1}{152}a^{18}-\frac{5}{152}a^{17}-\frac{5}{152}a^{16}-\frac{7}{76}a^{15}+\frac{1}{19}a^{14}+\frac{4}{19}a^{13}-\frac{3}{19}a^{12}+\frac{3}{38}a^{11}-\frac{7}{76}a^{10}-\frac{5}{76}a^{9}+\frac{13}{76}a^{8}-\frac{9}{19}a^{7}+\frac{75}{152}a^{6}+\frac{41}{152}a^{5}+\frac{47}{152}a^{4}+\frac{5}{38}a^{3}-\frac{11}{76}a^{2}+\frac{8}{19}a-\frac{15}{38}$, $\frac{1}{13\cdots 08}a^{19}+\frac{28\cdots 41}{13\cdots 08}a^{18}-\frac{21\cdots 27}{13\cdots 08}a^{17}-\frac{25\cdots 63}{22\cdots 68}a^{16}-\frac{27\cdots 05}{33\cdots 02}a^{15}+\frac{20\cdots 74}{16\cdots 01}a^{14}-\frac{46\cdots 29}{33\cdots 02}a^{13}+\frac{59\cdots 75}{33\cdots 02}a^{12}+\frac{13\cdots 13}{66\cdots 04}a^{11}-\frac{54\cdots 57}{66\cdots 04}a^{10}+\frac{55\cdots 75}{22\cdots 68}a^{9}+\frac{49\cdots 17}{33\cdots 02}a^{8}-\frac{48\cdots 57}{13\cdots 08}a^{7}+\frac{22\cdots 03}{13\cdots 08}a^{6}+\frac{42\cdots 49}{13\cdots 08}a^{5}-\frac{27\cdots 05}{16\cdots 01}a^{4}-\frac{70\cdots 49}{66\cdots 04}a^{3}-\frac{75\cdots 55}{37\cdots 78}a^{2}-\frac{36\cdots 59}{11\cdots 34}a+\frac{20\cdots 68}{87\cdots 79}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Ideal class group:		$C_{4}$, which has order $4$ (assuming GRH)
Narrow class group:		$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{30\cdots 07}{22\cdots 88}a^{19}+\frac{62\cdots 22}{56\cdots 47}a^{18}-\frac{16\cdots 20}{56\cdots 47}a^{17}+\frac{74\cdots 89}{56\cdots 47}a^{16}+\frac{23\cdots 13}{11\cdots 94}a^{15}-\frac{35\cdots 87}{22\cdots 88}a^{14}-\frac{36\cdots 66}{56\cdots 47}a^{13}+\frac{11\cdots 29}{22\cdots 88}a^{12}+\frac{11\cdots 93}{11\cdots 94}a^{11}-\frac{17\cdots 15}{22\cdots 88}a^{10}-\frac{50\cdots 58}{56\cdots 47}a^{9}+\frac{12\cdots 49}{22\cdots 88}a^{8}+\frac{90\cdots 45}{22\cdots 88}a^{7}-\frac{36\cdots 39}{22\cdots 88}a^{6}-\frac{10\cdots 61}{11\cdots 94}a^{5}+\frac{46\cdots 33}{22\cdots 88}a^{4}+\frac{10\cdots 75}{11\cdots 94}a^{3}-\frac{61\cdots 48}{56\cdots 47}a^{2}-\frac{16\cdots 27}{56\cdots 47}a+\frac{14\cdots 96}{56\cdots 47}$, $\frac{30\cdots 45}{22\cdots 88}a^{19}-\frac{30\cdots 59}{11\cdots 94}a^{18}-\frac{32\cdots 67}{11\cdots 94}a^{17}+\frac{10\cdots 41}{11\cdots 94}a^{16}+\frac{21\cdots 49}{11\cdots 94}a^{15}-\frac{16\cdots 43}{22\cdots 88}a^{14}-\frac{29\cdots 53}{56\cdots 47}a^{13}+\frac{48\cdots 59}{22\cdots 88}a^{12}+\frac{70\cdots 99}{11\cdots 94}a^{11}-\frac{69\cdots 19}{22\cdots 88}a^{10}-\frac{32\cdots 39}{11\cdots 94}a^{9}+\frac{46\cdots 61}{22\cdots 88}a^{8}+\frac{78\cdots 87}{22\cdots 88}a^{7}-\frac{14\cdots 11}{22\cdots 88}a^{6}+\frac{15\cdots 64}{56\cdots 47}a^{5}+\frac{17\cdots 39}{22\cdots 88}a^{4}-\frac{28\cdots 93}{56\cdots 47}a^{3}-\frac{16\cdots 86}{56\cdots 47}a^{2}+\frac{12\cdots 71}{56\cdots 47}a-\frac{94\cdots 18}{56\cdots 47}$, $\frac{19\cdots 83}{11\cdots 94}a^{19}-\frac{60\cdots 33}{22\cdots 88}a^{18}-\frac{20\cdots 52}{56\cdots 47}a^{17}+\frac{23\cdots 23}{22\cdots 88}a^{16}+\frac{55\cdots 99}{22\cdots 88}a^{15}-\frac{18\cdots 41}{22\cdots 88}a^{14}-\frac{15\cdots 05}{22\cdots 88}a^{13}+\frac{55\cdots 41}{22\cdots 88}a^{12}+\frac{20\cdots 77}{22\cdots 88}a^{11}-\frac{79\cdots 43}{22\cdots 88}a^{10}-\frac{12\cdots 61}{22\cdots 88}a^{9}+\frac{54\cdots 55}{22\cdots 88}a^{8}+\frac{28\cdots 01}{22\cdots 88}a^{7}-\frac{87\cdots 13}{11\cdots 94}a^{6}-\frac{12\cdots 67}{22\cdots 88}a^{5}+\frac{12\cdots 81}{11\cdots 94}a^{4}-\frac{34\cdots 59}{56\cdots 47}a^{3}-\frac{29\cdots 29}{56\cdots 47}a^{2}-\frac{19\cdots 58}{56\cdots 47}a+\frac{65\cdots 09}{56\cdots 47}$, $\frac{57\cdots 83}{22\cdots 88}a^{19}-\frac{31\cdots 81}{56\cdots 47}a^{18}-\frac{30\cdots 47}{56\cdots 47}a^{17}+\frac{10\cdots 52}{56\cdots 47}a^{16}+\frac{40\cdots 85}{11\cdots 94}a^{15}-\frac{32\cdots 99}{22\cdots 88}a^{14}-\frac{54\cdots 40}{56\cdots 47}a^{13}+\frac{96\cdots 89}{22\cdots 88}a^{12}+\frac{12\cdots 05}{11\cdots 94}a^{11}-\frac{13\cdots 23}{22\cdots 88}a^{10}-\frac{27\cdots 81}{56\cdots 47}a^{9}+\frac{91\cdots 73}{22\cdots 88}a^{8}-\frac{88\cdots 71}{22\cdots 88}a^{7}-\frac{27\cdots 83}{22\cdots 88}a^{6}+\frac{73\cdots 17}{11\cdots 94}a^{5}+\frac{34\cdots 45}{22\cdots 88}a^{4}-\frac{12\cdots 47}{11\cdots 94}a^{3}-\frac{33\cdots 24}{56\cdots 47}a^{2}+\frac{26\cdots 69}{56\cdots 47}a-\frac{21\cdots 26}{56\cdots 47}$, $\frac{10\cdots 91}{21\cdots 86}a^{19}-\frac{49\cdots 39}{42\cdots 72}a^{18}-\frac{11\cdots 45}{10\cdots 93}a^{17}+\frac{82\cdots 49}{21\cdots 86}a^{16}+\frac{72\cdots 65}{10\cdots 93}a^{15}-\frac{12\cdots 43}{42\cdots 72}a^{14}-\frac{19\cdots 77}{10\cdots 93}a^{13}+\frac{36\cdots 93}{42\cdots 72}a^{12}+\frac{22\cdots 70}{10\cdots 93}a^{11}-\frac{51\cdots 89}{42\cdots 72}a^{10}-\frac{84\cdots 86}{10\cdots 93}a^{9}+\frac{34\cdots 45}{42\cdots 72}a^{8}-\frac{34\cdots 39}{21\cdots 86}a^{7}-\frac{26\cdots 26}{10\cdots 93}a^{6}+\frac{15\cdots 22}{10\cdots 93}a^{5}+\frac{13\cdots 09}{42\cdots 72}a^{4}-\frac{25\cdots 22}{10\cdots 93}a^{3}-\frac{12\cdots 50}{10\cdots 93}a^{2}+\frac{10\cdots 00}{10\cdots 93}a-\frac{86\cdots 27}{10\cdots 93}$, $\frac{33\cdots 87}{13\cdots 08}a^{19}-\frac{24\cdots 51}{33\cdots 02}a^{18}-\frac{17\cdots 35}{33\cdots 02}a^{17}+\frac{10\cdots 85}{44\cdots 36}a^{16}+\frac{22\cdots 43}{66\cdots 04}a^{15}-\frac{56\cdots 95}{33\cdots 02}a^{14}-\frac{14\cdots 08}{16\cdots 01}a^{13}+\frac{84\cdots 09}{16\cdots 01}a^{12}+\frac{62\cdots 05}{66\cdots 04}a^{11}-\frac{23\cdots 57}{33\cdots 02}a^{10}-\frac{25\cdots 15}{11\cdots 34}a^{9}+\frac{32\cdots 27}{66\cdots 04}a^{8}-\frac{25\cdots 67}{13\cdots 08}a^{7}-\frac{25\cdots 75}{16\cdots 01}a^{6}+\frac{71\cdots 11}{66\cdots 04}a^{5}+\frac{27\cdots 33}{13\cdots 08}a^{4}-\frac{98\cdots 77}{66\cdots 04}a^{3}-\frac{66\cdots 19}{74\cdots 56}a^{2}+\frac{57\cdots 19}{11\cdots 34}a-\frac{11\cdots 23}{33\cdots 02}$, $\frac{82\cdots 29}{23\cdots 44}a^{19}-\frac{37\cdots 54}{55\cdots 67}a^{18}-\frac{16\cdots 77}{22\cdots 68}a^{17}+\frac{35\cdots 63}{14\cdots 12}a^{16}+\frac{27\cdots 68}{55\cdots 67}a^{15}-\frac{10\cdots 67}{55\cdots 67}a^{14}-\frac{30\cdots 13}{22\cdots 68}a^{13}+\frac{60\cdots 77}{11\cdots 34}a^{12}+\frac{18\cdots 53}{11\cdots 34}a^{11}-\frac{42\cdots 01}{55\cdots 67}a^{10}-\frac{59\cdots 01}{74\cdots 56}a^{9}+\frac{11\cdots 75}{22\cdots 68}a^{8}+\frac{17\cdots 47}{44\cdots 36}a^{7}-\frac{87\cdots 99}{55\cdots 67}a^{6}+\frac{13\cdots 31}{22\cdots 68}a^{5}+\frac{85\cdots 71}{44\cdots 36}a^{4}-\frac{25\cdots 79}{22\cdots 68}a^{3}-\frac{53\cdots 07}{74\cdots 56}a^{2}+\frac{19\cdots 89}{37\cdots 78}a-\frac{46\cdots 69}{11\cdots 34}$, $\frac{79\cdots 49}{13\cdots 08}a^{19}-\frac{28\cdots 28}{16\cdots 01}a^{18}-\frac{21\cdots 20}{16\cdots 01}a^{17}+\frac{23\cdots 71}{44\cdots 36}a^{16}+\frac{27\cdots 99}{33\cdots 02}a^{15}-\frac{26\cdots 91}{66\cdots 04}a^{14}-\frac{13\cdots 21}{66\cdots 04}a^{13}+\frac{78\cdots 03}{66\cdots 04}a^{12}+\frac{72\cdots 39}{33\cdots 02}a^{11}-\frac{10\cdots 27}{66\cdots 04}a^{10}-\frac{10\cdots 93}{22\cdots 68}a^{9}+\frac{18\cdots 72}{16\cdots 01}a^{8}-\frac{69\cdots 87}{13\cdots 08}a^{7}-\frac{22\cdots 51}{66\cdots 04}a^{6}+\frac{92\cdots 19}{33\cdots 02}a^{5}+\frac{54\cdots 21}{13\cdots 08}a^{4}-\frac{26\cdots 17}{66\cdots 04}a^{3}-\frac{10\cdots 57}{74\cdots 56}a^{2}+\frac{16\cdots 55}{11\cdots 34}a-\frac{41\cdots 07}{33\cdots 02}$, $\frac{99\cdots 83}{66\cdots 04}a^{19}-\frac{13\cdots 45}{33\cdots 02}a^{18}-\frac{10\cdots 97}{33\cdots 02}a^{17}+\frac{71\cdots 67}{55\cdots 67}a^{16}+\frac{13\cdots 15}{66\cdots 04}a^{15}-\frac{31\cdots 57}{33\cdots 02}a^{14}-\frac{35\cdots 03}{66\cdots 04}a^{13}+\frac{95\cdots 91}{33\cdots 02}a^{12}+\frac{38\cdots 61}{66\cdots 04}a^{11}-\frac{67\cdots 64}{16\cdots 01}a^{10}-\frac{37\cdots 71}{22\cdots 68}a^{9}+\frac{45\cdots 01}{16\cdots 01}a^{8}-\frac{33\cdots 33}{33\cdots 02}a^{7}-\frac{14\cdots 16}{16\cdots 01}a^{6}+\frac{40\cdots 03}{66\cdots 04}a^{5}+\frac{18\cdots 89}{16\cdots 01}a^{4}-\frac{15\cdots 25}{16\cdots 01}a^{3}-\frac{16\cdots 33}{37\cdots 78}a^{2}+\frac{21\cdots 01}{55\cdots 67}a-\frac{53\cdots 18}{16\cdots 01}$, $\frac{60\cdots 55}{22\cdots 68}a^{19}-\frac{54\cdots 43}{11\cdots 34}a^{18}-\frac{32\cdots 08}{55\cdots 67}a^{17}+\frac{17\cdots 57}{97\cdots 31}a^{16}+\frac{86\cdots 55}{22\cdots 68}a^{15}-\frac{75\cdots 49}{55\cdots 67}a^{14}-\frac{23\cdots 31}{22\cdots 68}a^{13}+\frac{45\cdots 79}{11\cdots 34}a^{12}+\frac{29\cdots 59}{22\cdots 68}a^{11}-\frac{64\cdots 65}{11\cdots 34}a^{10}-\frac{52\cdots 19}{74\cdots 56}a^{9}+\frac{43\cdots 41}{11\cdots 34}a^{8}+\frac{92\cdots 69}{11\cdots 34}a^{7}-\frac{67\cdots 47}{55\cdots 67}a^{6}+\frac{69\cdots 03}{22\cdots 68}a^{5}+\frac{17\cdots 65}{11\cdots 34}a^{4}-\frac{39\cdots 46}{55\cdots 67}a^{3}-\frac{24\cdots 47}{37\cdots 78}a^{2}+\frac{58\cdots 12}{18\cdots 89}a+\frac{11\cdots 07}{55\cdots 67}$, $\frac{38\cdots 53}{22\cdots 06}a^{19}-\frac{14\cdots 97}{17\cdots 56}a^{18}-\frac{62\cdots 87}{17\cdots 56}a^{17}+\frac{12\cdots 23}{57\cdots 52}a^{16}+\frac{48\cdots 38}{21\cdots 57}a^{15}-\frac{34\cdots 96}{21\cdots 57}a^{14}-\frac{46\cdots 91}{86\cdots 28}a^{13}+\frac{20\cdots 07}{43\cdots 14}a^{12}+\frac{39\cdots 47}{86\cdots 28}a^{11}-\frac{58\cdots 85}{86\cdots 28}a^{10}+\frac{14\cdots 99}{14\cdots 38}a^{9}+\frac{39\cdots 97}{86\cdots 28}a^{8}-\frac{25\cdots 29}{86\cdots 28}a^{7}-\frac{24\cdots 67}{17\cdots 56}a^{6}+\frac{20\cdots 01}{17\cdots 56}a^{5}+\frac{31\cdots 65}{17\cdots 56}a^{4}-\frac{70\cdots 75}{43\cdots 14}a^{3}-\frac{67\cdots 91}{95\cdots 92}a^{2}+\frac{45\cdots 60}{71\cdots 19}a-\frac{23\cdots 59}{43\cdots 14}$, $\frac{44\cdots 23}{14\cdots 12}a^{19}-\frac{43\cdots 55}{37\cdots 78}a^{18}-\frac{12\cdots 65}{19\cdots 62}a^{17}+\frac{16\cdots 35}{14\cdots 12}a^{16}+\frac{33\cdots 77}{74\cdots 56}a^{15}-\frac{69\cdots 61}{74\cdots 56}a^{14}-\frac{49\cdots 09}{37\cdots 78}a^{13}+\frac{23\cdots 37}{74\cdots 56}a^{12}+\frac{14\cdots 17}{74\cdots 56}a^{11}-\frac{36\cdots 61}{74\cdots 56}a^{10}-\frac{51\cdots 55}{37\cdots 78}a^{9}+\frac{15\cdots 95}{37\cdots 78}a^{8}+\frac{66\cdots 05}{14\cdots 12}a^{7}-\frac{12\cdots 99}{74\cdots 56}a^{6}-\frac{27\cdots 45}{74\cdots 56}a^{5}+\frac{48\cdots 03}{14\cdots 12}a^{4}-\frac{71\cdots 71}{74\cdots 56}a^{3}-\frac{15\cdots 13}{74\cdots 56}a^{2}+\frac{49\cdots 85}{37\cdots 78}a-\frac{63\cdots 29}{37\cdots 78}$, $\frac{24\cdots 38}{16\cdots 01}a^{19}-\frac{27\cdots 57}{66\cdots 04}a^{18}-\frac{52\cdots 11}{16\cdots 01}a^{17}+\frac{14\cdots 65}{11\cdots 34}a^{16}+\frac{33\cdots 70}{16\cdots 01}a^{15}-\frac{63\cdots 85}{66\cdots 04}a^{14}-\frac{17\cdots 41}{33\cdots 02}a^{13}+\frac{19\cdots 35}{66\cdots 04}a^{12}+\frac{92\cdots 65}{16\cdots 01}a^{11}-\frac{26\cdots 15}{66\cdots 04}a^{10}-\frac{78\cdots 79}{55\cdots 67}a^{9}+\frac{18\cdots 27}{66\cdots 04}a^{8}-\frac{37\cdots 85}{33\cdots 02}a^{7}-\frac{73\cdots 72}{87\cdots 79}a^{6}+\frac{10\cdots 61}{16\cdots 01}a^{5}+\frac{70\cdots 83}{66\cdots 04}a^{4}-\frac{83\cdots 76}{87\cdots 79}a^{3}-\frac{76\cdots 62}{18\cdots 89}a^{2}+\frac{20\cdots 48}{55\cdots 67}a-\frac{51\cdots 57}{16\cdots 01}$, $\frac{19\cdots 39}{13\cdots 08}a^{19}-\frac{22\cdots 11}{66\cdots 04}a^{18}-\frac{20\cdots 25}{66\cdots 04}a^{17}+\frac{50\cdots 67}{44\cdots 36}a^{16}+\frac{66\cdots 47}{33\cdots 02}a^{15}-\frac{14\cdots 97}{16\cdots 01}a^{14}-\frac{35\cdots 57}{66\cdots 04}a^{13}+\frac{42\cdots 97}{16\cdots 01}a^{12}+\frac{10\cdots 67}{16\cdots 01}a^{11}-\frac{31\cdots 94}{87\cdots 79}a^{10}-\frac{50\cdots 99}{22\cdots 68}a^{9}+\frac{15\cdots 77}{66\cdots 04}a^{8}-\frac{68\cdots 93}{13\cdots 08}a^{7}-\frac{48\cdots 15}{66\cdots 04}a^{6}+\frac{30\cdots 81}{66\cdots 04}a^{5}+\frac{63\cdots 57}{70\cdots 32}a^{4}-\frac{47\cdots 03}{66\cdots 04}a^{3}-\frac{25\cdots 99}{74\cdots 56}a^{2}+\frac{32\cdots 97}{11\cdots 34}a-\frac{81\cdots 13}{33\cdots 02}$, $\frac{25\cdots 91}{44\cdots 36}a^{19}-\frac{29\cdots 55}{22\cdots 68}a^{18}-\frac{27\cdots 97}{22\cdots 68}a^{17}+\frac{65\cdots 19}{14\cdots 12}a^{16}+\frac{44\cdots 55}{55\cdots 67}a^{15}-\frac{36\cdots 95}{11\cdots 34}a^{14}-\frac{47\cdots 81}{22\cdots 68}a^{13}+\frac{54\cdots 85}{55\cdots 67}a^{12}+\frac{13\cdots 65}{55\cdots 67}a^{11}-\frac{40\cdots 20}{29\cdots 93}a^{10}-\frac{71\cdots 59}{74\cdots 56}a^{9}+\frac{20\cdots 23}{22\cdots 68}a^{8}-\frac{71\cdots 45}{44\cdots 36}a^{7}-\frac{63\cdots 15}{22\cdots 68}a^{6}+\frac{36\cdots 61}{22\cdots 68}a^{5}+\frac{82\cdots 49}{23\cdots 44}a^{4}-\frac{59\cdots 01}{22\cdots 68}a^{3}-\frac{98\cdots 37}{74\cdots 56}a^{2}+\frac{41\cdots 59}{37\cdots 78}a-\frac{10\cdots 13}{11\cdots 34}$, $\frac{92\cdots 69}{44\cdots 36}a^{19}-\frac{43\cdots 69}{11\cdots 34}a^{18}-\frac{99\cdots 83}{22\cdots 68}a^{17}+\frac{21\cdots 41}{14\cdots 12}a^{16}+\frac{64\cdots 11}{22\cdots 68}a^{15}-\frac{23\cdots 83}{22\cdots 68}a^{14}-\frac{88\cdots 59}{11\cdots 34}a^{13}+\frac{70\cdots 47}{22\cdots 68}a^{12}+\frac{21\cdots 33}{22\cdots 68}a^{11}-\frac{97\cdots 49}{22\cdots 68}a^{10}-\frac{17\cdots 65}{37\cdots 78}a^{9}+\frac{15\cdots 60}{55\cdots 67}a^{8}+\frac{16\cdots 39}{44\cdots 36}a^{7}-\frac{19\cdots 45}{22\cdots 68}a^{6}+\frac{16\cdots 11}{55\cdots 67}a^{5}+\frac{47\cdots 75}{44\cdots 36}a^{4}-\frac{13\cdots 97}{22\cdots 68}a^{3}-\frac{29\cdots 83}{74\cdots 56}a^{2}+\frac{10\cdots 35}{37\cdots 78}a-\frac{15\cdots 91}{58\cdots 86}$, $\frac{67\cdots 77}{11\cdots 34}a^{19}-\frac{74\cdots 51}{44\cdots 36}a^{18}-\frac{57\cdots 19}{44\cdots 36}a^{17}+\frac{78\cdots 89}{14\cdots 12}a^{16}+\frac{18\cdots 71}{22\cdots 68}a^{15}-\frac{86\cdots 15}{22\cdots 68}a^{14}-\frac{24\cdots 31}{11\cdots 34}a^{13}+\frac{13\cdots 01}{11\cdots 72}a^{12}+\frac{25\cdots 83}{11\cdots 34}a^{11}-\frac{91\cdots 42}{55\cdots 67}a^{10}-\frac{45\cdots 83}{74\cdots 56}a^{9}+\frac{61\cdots 86}{55\cdots 67}a^{8}-\frac{24\cdots 80}{55\cdots 67}a^{7}-\frac{14\cdots 47}{44\cdots 36}a^{6}+\frac{11\cdots 95}{44\cdots 36}a^{5}+\frac{18\cdots 25}{44\cdots 36}a^{4}-\frac{21\cdots 63}{55\cdots 67}a^{3}-\frac{11\cdots 27}{74\cdots 56}a^{2}+\frac{28\cdots 32}{18\cdots 89}a-\frac{14\cdots 67}{11\cdots 34}$, $\frac{67\cdots 23}{55\cdots 67}a^{19}-\frac{74\cdots 79}{22\cdots 68}a^{18}-\frac{57\cdots 09}{22\cdots 68}a^{17}+\frac{19\cdots 18}{18\cdots 89}a^{16}+\frac{36\cdots 07}{22\cdots 68}a^{15}-\frac{17\cdots 49}{22\cdots 68}a^{14}-\frac{96\cdots 01}{22\cdots 68}a^{13}+\frac{51\cdots 45}{22\cdots 68}a^{12}+\frac{10\cdots 67}{22\cdots 68}a^{11}-\frac{72\cdots 79}{22\cdots 68}a^{10}-\frac{89\cdots 53}{74\cdots 56}a^{9}+\frac{48\cdots 43}{22\cdots 68}a^{8}-\frac{20\cdots 69}{22\cdots 68}a^{7}-\frac{37\cdots 40}{55\cdots 67}a^{6}+\frac{28\cdots 10}{55\cdots 67}a^{5}+\frac{18\cdots 09}{22\cdots 68}a^{4}-\frac{84\cdots 45}{11\cdots 34}a^{3}-\frac{56\cdots 40}{18\cdots 89}a^{2}+\frac{55\cdots 59}{18\cdots 89}a-\frac{16\cdots 47}{55\cdots 67}$, $\frac{10\cdots 01}{42\cdots 72}a^{19}-\frac{14\cdots 51}{21\cdots 86}a^{18}-\frac{22\cdots 41}{42\cdots 72}a^{17}+\frac{23\cdots 46}{10\cdots 93}a^{16}+\frac{14\cdots 43}{42\cdots 72}a^{15}-\frac{34\cdots 61}{21\cdots 86}a^{14}-\frac{38\cdots 77}{42\cdots 72}a^{13}+\frac{10\cdots 29}{21\cdots 86}a^{12}+\frac{41\cdots 81}{42\cdots 72}a^{11}-\frac{71\cdots 09}{10\cdots 93}a^{10}-\frac{11\cdots 63}{42\cdots 72}a^{9}+\frac{47\cdots 99}{10\cdots 93}a^{8}-\frac{37\cdots 61}{21\cdots 86}a^{7}-\frac{14\cdots 81}{10\cdots 93}a^{6}+\frac{21\cdots 27}{21\cdots 86}a^{5}+\frac{18\cdots 89}{10\cdots 93}a^{4}-\frac{31\cdots 41}{21\cdots 86}a^{3}-\frac{13\cdots 45}{21\cdots 86}a^{2}+\frac{62\cdots 82}{10\cdots 93}a-\frac{51\cdots 51}{10\cdots 93}$ (assuming GRH)
Regulator:		\( 1146610929550000000 \) (assuming GRH)
Unit signature rank:		\( 16 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 1146610929550000000 \cdot 4}{2\cdot\sqrt{258866544008047715925279948537661096282306445312}}\cr\approx \mathstrut & 4.72615568034128 \end{aligned}\] (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

\(\Q(\sqrt{901}) \), \(\Q(\sqrt{102 +2 \sqrt{901}})\), 5.5.2382032.1, 10.10.426987989728139087104.1

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:	20.20.4884274415246183319344904689389832005326536704.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$	${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$	$20$	$20$	${\href{/padicField/13.10.0.1}{10} }^{2}$	R	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.4.0.1}{4} }^{5}$	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.10.0.1}{10} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$	R	${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(2\)	2.2.1.0a1.1	$x^{2} + x + 1$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	2.2.1.0a1.1	$x^{2} + x + 1$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	2.2.2.4a1.2	$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$	$2$	$2$	$4$	$C_4$	$$[2]^{2}$$
	2.2.2.4a1.2	$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$	$2$	$2$	$4$	$C_4$	$$[2]^{2}$$
	2.2.2.4a1.2	$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$	$2$	$2$	$4$	$C_4$	$$[2]^{2}$$
	2.2.2.4a1.2	$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$	$2$	$2$	$4$	$C_4$	$$[2]^{2}$$
\(17\)	17.1.4.3a1.3	$x^{4} + 153$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$
	17.2.4.6a1.2	$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$	$4$	$2$	$6$	$C_4\times C_2$	$$[\ ]_{4}^{2}$$
	17.2.4.6a1.2	$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$	$4$	$2$	$6$	$C_4\times C_2$	$$[\ ]_{4}^{2}$$
\(53\)	53.1.2.1a1.1	$x^{2} + 53$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	53.1.2.1a1.1	$x^{2} + 53$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	53.1.4.3a1.1	$x^{4} + 53$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$
	53.1.4.3a1.1	$x^{4} + 53$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$
	53.1.4.3a1.1	$x^{4} + 53$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$
	53.1.4.3a1.1	$x^{4} + 53$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$

Spectrum of ring of integers