Properties

Label 18.6.950...000.1
Degree 1818
Signature [6,6][6, 6]
Discriminant 9.506×10349.506\times 10^{34}
Root discriminant 87.7487.74
Ramified primes 2,3,52,3,5
Class number 11 (GRH)
Class group trivial (GRH)
Galois group C36:C23:S4C_3^6:C_2^3:S_4 (as 18T822)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088)
 
Copy content gp:K = bnfinit(y^18 - 36*y^16 - 24*y^15 + 432*y^14 + 576*y^13 - 2088*y^12 - 3888*y^11 + 4284*y^10 + 10240*y^9 - 5040*y^7 - 13530*y^6 - 32832*y^5 + 9576*y^4 + 63360*y^3 + 7128*y^2 - 1152*y - 1088, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088)
 

x1836x1624x15+432x14+576x132088x123888x11+4284x10+1088 x^{18} - 36 x^{16} - 24 x^{15} + 432 x^{14} + 576 x^{13} - 2088 x^{12} - 3888 x^{11} + 4284 x^{10} + \cdots - 1088 Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  1818
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  [6,6][6, 6]
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   9505773894288877559352068250009600095057738942888775593520682500096000 =24933853\medspace = 2^{49}\cdot 3^{38}\cdot 5^{3} Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  87.7487.74
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   22, 33, 55 Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q(10)\Q(\sqrt{10})
Aut(K/Q)\Aut(K/\Q):   C1C_1
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, a8a^{8}, 13a913\frac{1}{3}a^{9}-\frac{1}{3}, 13a1013a\frac{1}{3}a^{10}-\frac{1}{3}a, 13a1113a2\frac{1}{3}a^{11}-\frac{1}{3}a^{2}, 118a12+19a913a6+49a3+29\frac{1}{18}a^{12}+\frac{1}{9}a^{9}-\frac{1}{3}a^{6}+\frac{4}{9}a^{3}+\frac{2}{9}, 118a13+19a1013a7+49a4+29a\frac{1}{18}a^{13}+\frac{1}{9}a^{10}-\frac{1}{3}a^{7}+\frac{4}{9}a^{4}+\frac{2}{9}a, 136a1419a11+13a8+29a5+518a2\frac{1}{36}a^{14}-\frac{1}{9}a^{11}+\frac{1}{3}a^{8}+\frac{2}{9}a^{5}+\frac{5}{18}a^{2}, 172a15136a1316a11+19a10+19a9+16a729a629a4+112a313a2518a19\frac{1}{72}a^{15}-\frac{1}{36}a^{13}-\frac{1}{6}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{6}a^{7}-\frac{2}{9}a^{6}-\frac{2}{9}a^{4}+\frac{1}{12}a^{3}-\frac{1}{3}a^{2}-\frac{5}{18}a-\frac{1}{9}, 1288a16+1144a14136a13172a12+118a1119a10+118a9+524a8+19a7512a6+118a5+43144a419a3+3572a2+49a+19\frac{1}{288}a^{16}+\frac{1}{144}a^{14}-\frac{1}{36}a^{13}-\frac{1}{72}a^{12}+\frac{1}{18}a^{11}-\frac{1}{9}a^{10}+\frac{1}{18}a^{9}+\frac{5}{24}a^{8}+\frac{1}{9}a^{7}-\frac{5}{12}a^{6}+\frac{1}{18}a^{5}+\frac{43}{144}a^{4}-\frac{1}{9}a^{3}+\frac{35}{72}a^{2}+\frac{4}{9}a+\frac{1}{9}, 15164a1765395164a16+52032532a1544818544a14+31191216a13+42311216a1219581627a1115631018a1055074272a959891216a8+14636408a7+79036408a610812848a570972532a4+25971216a361971216a2+40071018a16681703\frac{1}{51\cdots 64}a^{17}-\frac{65\cdots 39}{51\cdots 64}a^{16}+\frac{52\cdots 03}{25\cdots 32}a^{15}-\frac{44\cdots 81}{85\cdots 44}a^{14}+\frac{31\cdots 19}{12\cdots 16}a^{13}+\frac{42\cdots 31}{12\cdots 16}a^{12}-\frac{19\cdots 58}{16\cdots 27}a^{11}-\frac{15\cdots 63}{10\cdots 18}a^{10}-\frac{55\cdots 07}{42\cdots 72}a^{9}-\frac{59\cdots 89}{12\cdots 16}a^{8}+\frac{14\cdots 63}{64\cdots 08}a^{7}+\frac{79\cdots 03}{64\cdots 08}a^{6}-\frac{10\cdots 81}{28\cdots 48}a^{5}-\frac{70\cdots 97}{25\cdots 32}a^{4}+\frac{25\cdots 97}{12\cdots 16}a^{3}-\frac{61\cdots 97}{12\cdots 16}a^{2}+\frac{40\cdots 07}{10\cdots 18}a-\frac{16\cdots 68}{17\cdots 03} Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  22

Class group and class number

Ideal class group:  Trivial group, which has order 11 (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order 11 (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  1111
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   1 -1  (order 22) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   49701703a17+90105309a1615481627a1521371703a14+18211703a13+11975309a1268141703a1159065309a10+62621627a9+40761703a8+57975309a7+42771627a646921703a516001703a428151703a3+20401703a2+63225309a+10371627\frac{49\cdots 70}{17\cdots 03}a^{17}+\frac{90\cdots 10}{53\cdots 09}a^{16}-\frac{15\cdots 48}{16\cdots 27}a^{15}-\frac{21\cdots 37}{17\cdots 03}a^{14}+\frac{18\cdots 21}{17\cdots 03}a^{13}+\frac{11\cdots 97}{53\cdots 09}a^{12}-\frac{68\cdots 14}{17\cdots 03}a^{11}-\frac{59\cdots 06}{53\cdots 09}a^{10}+\frac{62\cdots 62}{16\cdots 27}a^{9}+\frac{40\cdots 76}{17\cdots 03}a^{8}+\frac{57\cdots 97}{53\cdots 09}a^{7}+\frac{42\cdots 77}{16\cdots 27}a^{6}-\frac{46\cdots 92}{17\cdots 03}a^{5}-\frac{16\cdots 00}{17\cdots 03}a^{4}-\frac{28\cdots 15}{17\cdots 03}a^{3}+\frac{20\cdots 40}{17\cdots 03}a^{2}+\frac{63\cdots 22}{53\cdots 09}a+\frac{10\cdots 37}{16\cdots 27}, 19661703a1760145309a1664401627a1540171703a14+87191703a13+31395309a1243981703a1121785309a10+87581627a9+19361703a889575309a795711627a625961703a563201703a4+85055309a3+12601703a263345309a65111627\frac{19\cdots 66}{17\cdots 03}a^{17}-\frac{60\cdots 14}{53\cdots 09}a^{16}-\frac{64\cdots 40}{16\cdots 27}a^{15}-\frac{40\cdots 17}{17\cdots 03}a^{14}+\frac{87\cdots 19}{17\cdots 03}a^{13}+\frac{31\cdots 39}{53\cdots 09}a^{12}-\frac{43\cdots 98}{17\cdots 03}a^{11}-\frac{21\cdots 78}{53\cdots 09}a^{10}+\frac{87\cdots 58}{16\cdots 27}a^{9}+\frac{19\cdots 36}{17\cdots 03}a^{8}-\frac{89\cdots 57}{53\cdots 09}a^{7}-\frac{95\cdots 71}{16\cdots 27}a^{6}-\frac{25\cdots 96}{17\cdots 03}a^{5}-\frac{63\cdots 20}{17\cdots 03}a^{4}+\frac{85\cdots 05}{53\cdots 09}a^{3}+\frac{12\cdots 60}{17\cdots 03}a^{2}-\frac{63\cdots 34}{53\cdots 09}a-\frac{65\cdots 11}{16\cdots 27}, 31321703a17+36981703a1612972136a1590441703a14+22113506a13+16191627a1240911703a1129005309a10+35351627a9+17321703a8+13661703a7+24485309a629161703a591521703a412353254a3+75921703a2+77375309a+50531627\frac{31\cdots 32}{17\cdots 03}a^{17}+\frac{36\cdots 98}{17\cdots 03}a^{16}-\frac{12\cdots 97}{21\cdots 36}a^{15}-\frac{90\cdots 44}{17\cdots 03}a^{14}+\frac{22\cdots 11}{35\cdots 06}a^{13}+\frac{16\cdots 19}{16\cdots 27}a^{12}-\frac{40\cdots 91}{17\cdots 03}a^{11}-\frac{29\cdots 00}{53\cdots 09}a^{10}+\frac{35\cdots 35}{16\cdots 27}a^{9}+\frac{17\cdots 32}{17\cdots 03}a^{8}+\frac{13\cdots 66}{17\cdots 03}a^{7}+\frac{24\cdots 48}{53\cdots 09}a^{6}-\frac{29\cdots 16}{17\cdots 03}a^{5}-\frac{91\cdots 52}{17\cdots 03}a^{4}-\frac{12\cdots 35}{32\cdots 54}a^{3}+\frac{75\cdots 92}{17\cdots 03}a^{2}+\frac{77\cdots 37}{53\cdots 09}a+\frac{50\cdots 53}{16\cdots 27}, 81475164a17+16191788a1614592532a1518312532a14+26834272a13+15831216a1243811703a1173231018a10+37531216a9+19631216a8+13992136a712416408a643032532a546118544a415771216a3+33334272a2+30651018a+90461627\frac{81\cdots 47}{51\cdots 64}a^{17}+\frac{16\cdots 19}{17\cdots 88}a^{16}-\frac{14\cdots 59}{25\cdots 32}a^{15}-\frac{18\cdots 31}{25\cdots 32}a^{14}+\frac{26\cdots 83}{42\cdots 72}a^{13}+\frac{15\cdots 83}{12\cdots 16}a^{12}-\frac{43\cdots 81}{17\cdots 03}a^{11}-\frac{73\cdots 23}{10\cdots 18}a^{10}+\frac{37\cdots 53}{12\cdots 16}a^{9}+\frac{19\cdots 63}{12\cdots 16}a^{8}+\frac{13\cdots 99}{21\cdots 36}a^{7}-\frac{12\cdots 41}{64\cdots 08}a^{6}-\frac{43\cdots 03}{25\cdots 32}a^{5}-\frac{46\cdots 11}{85\cdots 44}a^{4}-\frac{15\cdots 77}{12\cdots 16}a^{3}+\frac{33\cdots 33}{42\cdots 72}a^{2}+\frac{30\cdots 65}{10\cdots 18}a+\frac{90\cdots 46}{16\cdots 27}, 36771216a1773231627a1665016408a15+15441627a14+21641627a1351741627a1214941627a1131551627a10+98793254a9+19751627a876511627a730171627a611176408a552991627a4+58073254a3+24621627a241901627a86271627\frac{36\cdots 77}{12\cdots 16}a^{17}-\frac{73\cdots 23}{16\cdots 27}a^{16}-\frac{65\cdots 01}{64\cdots 08}a^{15}+\frac{15\cdots 44}{16\cdots 27}a^{14}+\frac{21\cdots 64}{16\cdots 27}a^{13}-\frac{51\cdots 74}{16\cdots 27}a^{12}-\frac{14\cdots 94}{16\cdots 27}a^{11}-\frac{31\cdots 55}{16\cdots 27}a^{10}+\frac{98\cdots 79}{32\cdots 54}a^{9}+\frac{19\cdots 75}{16\cdots 27}a^{8}-\frac{76\cdots 51}{16\cdots 27}a^{7}-\frac{30\cdots 17}{16\cdots 27}a^{6}-\frac{11\cdots 17}{64\cdots 08}a^{5}-\frac{52\cdots 99}{16\cdots 27}a^{4}+\frac{58\cdots 07}{32\cdots 54}a^{3}+\frac{24\cdots 62}{16\cdots 27}a^{2}-\frac{41\cdots 90}{16\cdots 27}a-\frac{86\cdots 27}{16\cdots 27}, 98874272a17+46975164a1610791216a1522312532a14+31713254a13+22551216a1272681627a1118641627a10+11691627a9+42911424a8+23053254a711956408a623596408a522052532a418376408a3+22711216a2+23753254a45141627\frac{98\cdots 87}{42\cdots 72}a^{17}+\frac{46\cdots 97}{51\cdots 64}a^{16}-\frac{10\cdots 79}{12\cdots 16}a^{15}-\frac{22\cdots 31}{25\cdots 32}a^{14}+\frac{31\cdots 71}{32\cdots 54}a^{13}+\frac{22\cdots 55}{12\cdots 16}a^{12}-\frac{72\cdots 68}{16\cdots 27}a^{11}-\frac{18\cdots 64}{16\cdots 27}a^{10}+\frac{11\cdots 69}{16\cdots 27}a^{9}+\frac{42\cdots 91}{14\cdots 24}a^{8}+\frac{23\cdots 05}{32\cdots 54}a^{7}-\frac{11\cdots 95}{64\cdots 08}a^{6}-\frac{23\cdots 59}{64\cdots 08}a^{5}-\frac{22\cdots 05}{25\cdots 32}a^{4}-\frac{18\cdots 37}{64\cdots 08}a^{3}+\frac{22\cdots 71}{12\cdots 16}a^{2}+\frac{23\cdots 75}{32\cdots 54}a-\frac{45\cdots 14}{16\cdots 27}, 70531788a1735695164a1643192532a15+93338544a14+33131216a13+15691216a1232681703a1190573506a10+56851216a9+44194272a8+19976408a738756408a615658544a586672532a430251216a3+63651424a2+17711018a+50101627\frac{70\cdots 53}{17\cdots 88}a^{17}-\frac{35\cdots 69}{51\cdots 64}a^{16}-\frac{43\cdots 19}{25\cdots 32}a^{15}+\frac{93\cdots 33}{85\cdots 44}a^{14}+\frac{33\cdots 13}{12\cdots 16}a^{13}+\frac{15\cdots 69}{12\cdots 16}a^{12}-\frac{32\cdots 68}{17\cdots 03}a^{11}-\frac{90\cdots 57}{35\cdots 06}a^{10}+\frac{56\cdots 85}{12\cdots 16}a^{9}+\frac{44\cdots 19}{42\cdots 72}a^{8}+\frac{19\cdots 97}{64\cdots 08}a^{7}-\frac{38\cdots 75}{64\cdots 08}a^{6}-\frac{15\cdots 65}{85\cdots 44}a^{5}-\frac{86\cdots 67}{25\cdots 32}a^{4}-\frac{30\cdots 25}{12\cdots 16}a^{3}+\frac{63\cdots 65}{14\cdots 24}a^{2}+\frac{17\cdots 71}{10\cdots 18}a+\frac{50\cdots 10}{16\cdots 27}, 85131627a17+89115164a1624271216a1548652532a14+73653254a13+49251216a1256035309a1139891627a10+57233254a9+84971216a8+43213254a723096408a625933254a549112532a4+23436408a3+54291424a2+44633254a91921627\frac{85\cdots 13}{16\cdots 27}a^{17}+\frac{89\cdots 11}{51\cdots 64}a^{16}-\frac{24\cdots 27}{12\cdots 16}a^{15}-\frac{48\cdots 65}{25\cdots 32}a^{14}+\frac{73\cdots 65}{32\cdots 54}a^{13}+\frac{49\cdots 25}{12\cdots 16}a^{12}-\frac{56\cdots 03}{53\cdots 09}a^{11}-\frac{39\cdots 89}{16\cdots 27}a^{10}+\frac{57\cdots 23}{32\cdots 54}a^{9}+\frac{84\cdots 97}{12\cdots 16}a^{8}+\frac{43\cdots 21}{32\cdots 54}a^{7}-\frac{23\cdots 09}{64\cdots 08}a^{6}-\frac{25\cdots 93}{32\cdots 54}a^{5}-\frac{49\cdots 11}{25\cdots 32}a^{4}+\frac{23\cdots 43}{64\cdots 08}a^{3}+\frac{54\cdots 29}{14\cdots 24}a^{2}+\frac{44\cdots 63}{32\cdots 54}a-\frac{91\cdots 92}{16\cdots 27}, 30956408a17+13732532a1626691627a1537151216a14+29941627a13+30236408a1297281627a1143901627a1040021627a9+39956408a8+88971627a724573254a626873254a528491216a422101627a3+18876408a2+54001627a+16891627\frac{30\cdots 95}{64\cdots 08}a^{17}+\frac{13\cdots 73}{25\cdots 32}a^{16}-\frac{26\cdots 69}{16\cdots 27}a^{15}-\frac{37\cdots 15}{12\cdots 16}a^{14}+\frac{29\cdots 94}{16\cdots 27}a^{13}+\frac{30\cdots 23}{64\cdots 08}a^{12}-\frac{97\cdots 28}{16\cdots 27}a^{11}-\frac{43\cdots 90}{16\cdots 27}a^{10}-\frac{40\cdots 02}{16\cdots 27}a^{9}+\frac{39\cdots 95}{64\cdots 08}a^{8}+\frac{88\cdots 97}{16\cdots 27}a^{7}-\frac{24\cdots 57}{32\cdots 54}a^{6}-\frac{26\cdots 87}{32\cdots 54}a^{5}-\frac{28\cdots 49}{12\cdots 16}a^{4}-\frac{22\cdots 10}{16\cdots 27}a^{3}+\frac{18\cdots 87}{64\cdots 08}a^{2}+\frac{54\cdots 00}{16\cdots 27}a+\frac{16\cdots 89}{16\cdots 27}, 14191788a17+17571216a1669992532a1511331703a14+29711216a13+13611627a1230561703a1190833254a1018491216a9+33331018a8+22976408a7+14173254a6+19592848a512096408a430131216a3+33253506a246321627a+82871627\frac{14\cdots 19}{17\cdots 88}a^{17}+\frac{17\cdots 57}{12\cdots 16}a^{16}-\frac{69\cdots 99}{25\cdots 32}a^{15}-\frac{11\cdots 33}{17\cdots 03}a^{14}+\frac{29\cdots 71}{12\cdots 16}a^{13}+\frac{13\cdots 61}{16\cdots 27}a^{12}-\frac{30\cdots 56}{17\cdots 03}a^{11}-\frac{90\cdots 83}{32\cdots 54}a^{10}-\frac{18\cdots 49}{12\cdots 16}a^{9}+\frac{33\cdots 33}{10\cdots 18}a^{8}+\frac{22\cdots 97}{64\cdots 08}a^{7}+\frac{14\cdots 17}{32\cdots 54}a^{6}+\frac{19\cdots 59}{28\cdots 48}a^{5}-\frac{12\cdots 09}{64\cdots 08}a^{4}-\frac{30\cdots 13}{12\cdots 16}a^{3}+\frac{33\cdots 25}{35\cdots 06}a^{2}-\frac{46\cdots 32}{16\cdots 27}a+\frac{82\cdots 87}{16\cdots 27}, 47495309a17+71295696a1611614272a1553098544a14+39041703a13+35694272a12+17881703a1153961703a1039971018a9+82774272a8+31813506a7+20832136a6+41711018a543932848a490032136a318651424a284493506a37025309\frac{47\cdots 49}{53\cdots 09}a^{17}+\frac{71\cdots 29}{56\cdots 96}a^{16}-\frac{11\cdots 61}{42\cdots 72}a^{15}-\frac{53\cdots 09}{85\cdots 44}a^{14}+\frac{39\cdots 04}{17\cdots 03}a^{13}+\frac{35\cdots 69}{42\cdots 72}a^{12}+\frac{17\cdots 88}{17\cdots 03}a^{11}-\frac{53\cdots 96}{17\cdots 03}a^{10}-\frac{39\cdots 97}{10\cdots 18}a^{9}+\frac{82\cdots 77}{42\cdots 72}a^{8}+\frac{31\cdots 81}{35\cdots 06}a^{7}+\frac{20\cdots 83}{21\cdots 36}a^{6}+\frac{41\cdots 71}{10\cdots 18}a^{5}-\frac{43\cdots 93}{28\cdots 48}a^{4}-\frac{90\cdots 03}{21\cdots 36}a^{3}-\frac{18\cdots 65}{14\cdots 24}a^{2}-\frac{84\cdots 49}{35\cdots 06}a-\frac{37\cdots 02}{53\cdots 09} Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  484266827745 484266827745 (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(26(2π)64842668277451295057738942888775593520682500096000(3.09257449197299 \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^{6}\cdot(2\pi)^{6}\cdot 484266827745 \cdot 1}{2\cdot\sqrt{95057738942888775593520682500096000}}\cr\approx \mathstrut & 3.09257449197299 \end{aligned} (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 36*x^16 - 24*x^15 + 432*x^14 + 576*x^13 - 2088*x^12 - 3888*x^11 + 4284*x^10 + 10240*x^9 - 5040*x^7 - 13530*x^6 - 32832*x^5 + 9576*x^4 + 63360*x^3 + 7128*x^2 - 1152*x - 1088); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

C36:C23:S4C_3^6:C_2^3:S_4 (as 18T822):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 139968
The 93 conjugacy class representatives for C36:C23:S4C_3^6:C_2^3:S_4
Character table for C36:C23:S4C_3^6:C_2^3:S_4

Intermediate fields

3.1.216.1, 6.2.11943936.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 sibling: data not computed
Degree 24 sibling: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R R R 9,6,3{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} } 9,6,3{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} } 42,24,12{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2} 42,3,23,1{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} } 12,4,2{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} } 44,2{\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} } 42,3,23,1{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} } 9,33{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3} 12,23{\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3} 12,23{\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3} 6,43{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{3} 12,4,2{\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} } 9,33{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{3} 42,3,23,1{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.1.2.3a1.2x2+10x^{2} + 10221133C2C_2[3][3]
2.1.4.11a1.5x4+10x^{4} + 1044111111D4D_{4}[2,3,4][2, 3, 4]
2.1.12.35a1.354x12+4x6+8x3+26x^{12} + 4 x^{6} + 8 x^{3} + 26121211353512T28[2,3,4]32[2, 3, 4]_{3}^{2}
33 Copy content Toggle raw display 3.1.9.19b2.48x9+18x6+18x4+6x3+9x2+3x^{9} + 18 x^{6} + 18 x^{4} + 6 x^{3} + 9 x^{2} + 399111919C3S3C_3 \wr S_3 [32,2,52,83]2[\frac{3}{2}, 2, \frac{5}{2}, \frac{8}{3}]_{2}
3.1.9.19b2.37x9+9x4+6x3+9x2+3x^{9} + 9 x^{4} + 6 x^{3} + 9 x^{2} + 399111919C3S3C_3 \wr S_3 [32,2,52,83]2[\frac{3}{2}, 2, \frac{5}{2}, \frac{8}{3}]_{2}
55 Copy content Toggle raw display 5.3.1.0a1.1x3+3x+3x^{3} + 3 x + 3113300C3C_3[ ]3[\ ]^{3}
5.3.2.3a1.1x6+6x4+6x3+9x2+23x+9x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9223333C6C_6[ ]23[\ ]_{2}^{3}
5.9.1.0a1.1x9+2x3+x+3x^{9} + 2 x^{3} + x + 3119900C9C_9[ ]9[\ ]^{9}

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)