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Properties

Label 40.0.20484662179...8561.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 11^{36}\cdot 13^{20}$
Root discriminant $54.05$
Ramified primes $3, 11, 13$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3486784401, 1162261467, -1162261467, -1291401630, -430467210, 430467210, 607437063, 202479021, -202479021, -272806380, -90935460, -98749611, 60308712, 104407380, 29074707, -32440500, -42037380, -14425803, 14248656, 19436340, 6419731, -6478780, 1583184, 534289, -518980, 133500, 39883, -47740, 9192, 5017, -1540, 1540, -381, -127, 127, -30, -10, 10, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401)
 
gp: K = bnfinit(x^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} - 3 x^{38} + 10 x^{37} - 10 x^{36} - 30 x^{35} + 127 x^{34} - 127 x^{33} - 381 x^{32} + 1540 x^{31} - 1540 x^{30} + 5017 x^{29} + 9192 x^{28} - 47740 x^{27} + 39883 x^{26} + 133500 x^{25} - 518980 x^{24} + 534289 x^{23} + 1583184 x^{22} - 6478780 x^{21} + 6419731 x^{20} + 19436340 x^{19} + 14248656 x^{18} - 14425803 x^{17} - 42037380 x^{16} - 32440500 x^{15} + 29074707 x^{14} + 104407380 x^{13} + 60308712 x^{12} - 98749611 x^{11} - 90935460 x^{10} - 272806380 x^{9} - 202479021 x^{8} + 202479021 x^{7} + 607437063 x^{6} + 430467210 x^{5} - 430467210 x^{4} - 1291401630 x^{3} - 1162261467 x^{2} + 1162261467 x + 3486784401 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2048466217933115502043842255668249817786415413279145255704960577548561=3^{20}\cdot 11^{36}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.05$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(429=3\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{429}(1,·)$, $\chi_{429}(131,·)$, $\chi_{429}(389,·)$, $\chi_{429}(391,·)$, $\chi_{429}(142,·)$, $\chi_{429}(272,·)$, $\chi_{429}(274,·)$, $\chi_{429}(259,·)$, $\chi_{429}(404,·)$, $\chi_{429}(25,·)$, $\chi_{429}(155,·)$, $\chi_{429}(157,·)$, $\chi_{429}(415,·)$, $\chi_{429}(38,·)$, $\chi_{429}(40,·)$, $\chi_{429}(170,·)$, $\chi_{429}(428,·)$, $\chi_{429}(53,·)$, $\chi_{429}(311,·)$, $\chi_{429}(313,·)$, $\chi_{429}(287,·)$, $\chi_{429}(181,·)$, $\chi_{429}(64,·)$, $\chi_{429}(194,·)$, $\chi_{429}(196,·)$, $\chi_{429}(326,·)$, $\chi_{429}(79,·)$, $\chi_{429}(337,·)$, $\chi_{429}(14,·)$, $\chi_{429}(248,·)$, $\chi_{429}(92,·)$, $\chi_{429}(350,·)$, $\chi_{429}(103,·)$, $\chi_{429}(233,·)$, $\chi_{429}(235,·)$, $\chi_{429}(365,·)$, $\chi_{429}(116,·)$, $\chi_{429}(118,·)$, $\chi_{429}(376,·)$, $\chi_{429}(298,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{24147} a^{22} - \frac{1}{9} a^{21} - \frac{1}{3} a^{20} + \frac{1}{9} a^{19} - \frac{1}{9} a^{18} - \frac{1}{3} a^{17} + \frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{6160}{24147} a^{11} + \frac{1}{3} a^{10} - \frac{4}{9} a^{9} + \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{902}{2683}$, $\frac{1}{72441} a^{23} - \frac{1}{72441} a^{22} - \frac{1}{9} a^{21} + \frac{1}{27} a^{20} - \frac{1}{27} a^{19} - \frac{1}{9} a^{18} + \frac{10}{27} a^{17} - \frac{10}{27} a^{16} - \frac{1}{9} a^{15} - \frac{8}{27} a^{14} + \frac{8}{27} a^{13} - \frac{17987}{72441} a^{12} + \frac{4207}{24147} a^{11} + \frac{5}{27} a^{10} - \frac{5}{27} a^{9} + \frac{4}{9} a^{8} - \frac{4}{27} a^{7} + \frac{4}{27} a^{6} + \frac{4}{9} a^{5} - \frac{13}{27} a^{4} + \frac{13}{27} a^{3} + \frac{4}{9} a^{2} - \frac{1195}{2683} a + \frac{1195}{2683}$, $\frac{1}{217323} a^{24} - \frac{1}{217323} a^{23} - \frac{1}{72441} a^{22} + \frac{1}{81} a^{21} - \frac{1}{81} a^{20} - \frac{1}{27} a^{19} + \frac{37}{81} a^{18} - \frac{37}{81} a^{17} - \frac{10}{27} a^{16} - \frac{8}{81} a^{15} + \frac{8}{81} a^{14} - \frac{17987}{217323} a^{13} - \frac{19940}{72441} a^{12} - \frac{45310}{217323} a^{11} - \frac{5}{81} a^{10} - \frac{5}{27} a^{9} + \frac{23}{81} a^{8} - \frac{23}{81} a^{7} + \frac{4}{27} a^{6} - \frac{40}{81} a^{5} + \frac{40}{81} a^{4} + \frac{13}{27} a^{3} - \frac{3878}{8049} a^{2} + \frac{3878}{8049} a + \frac{1195}{2683}$, $\frac{1}{651969} a^{25} - \frac{1}{651969} a^{24} - \frac{1}{217323} a^{23} + \frac{10}{651969} a^{22} - \frac{1}{243} a^{21} - \frac{1}{81} a^{20} + \frac{37}{243} a^{19} - \frac{37}{243} a^{18} - \frac{37}{81} a^{17} - \frac{89}{243} a^{16} + \frac{89}{243} a^{15} - \frac{235310}{651969} a^{14} - \frac{19940}{217323} a^{13} - \frac{45310}{651969} a^{12} + \frac{37453}{651969} a^{11} - \frac{5}{81} a^{10} + \frac{104}{243} a^{9} - \frac{104}{243} a^{8} - \frac{23}{81} a^{7} - \frac{40}{243} a^{6} + \frac{40}{243} a^{5} + \frac{40}{81} a^{4} + \frac{4171}{24147} a^{3} - \frac{4171}{24147} a^{2} + \frac{3878}{8049} a + \frac{759}{2683}$, $\frac{1}{1955907} a^{26} - \frac{1}{1955907} a^{25} - \frac{1}{651969} a^{24} + \frac{10}{1955907} a^{23} - \frac{10}{1955907} a^{22} - \frac{1}{243} a^{21} + \frac{37}{729} a^{20} - \frac{37}{729} a^{19} - \frac{37}{243} a^{18} - \frac{332}{729} a^{17} + \frac{332}{729} a^{16} - \frac{887279}{1955907} a^{15} - \frac{237263}{651969} a^{14} - \frac{697279}{1955907} a^{13} + \frac{689422}{1955907} a^{12} + \frac{42070}{651969} a^{11} + \frac{347}{729} a^{10} - \frac{347}{729} a^{9} - \frac{104}{243} a^{8} - \frac{40}{729} a^{7} + \frac{40}{729} a^{6} + \frac{40}{243} a^{5} + \frac{4171}{72441} a^{4} - \frac{4171}{72441} a^{3} - \frac{4171}{24147} a^{2} + \frac{253}{2683} a - \frac{253}{2683}$, $\frac{1}{5867721} a^{27} - \frac{1}{5867721} a^{26} - \frac{1}{1955907} a^{25} + \frac{10}{5867721} a^{24} - \frac{10}{5867721} a^{23} - \frac{10}{1955907} a^{22} + \frac{37}{2187} a^{21} - \frac{37}{2187} a^{20} - \frac{37}{729} a^{19} + \frac{397}{2187} a^{18} - \frac{397}{2187} a^{17} - \frac{887279}{5867721} a^{16} - \frac{237263}{1955907} a^{15} - \frac{2653186}{5867721} a^{14} + \frac{689422}{5867721} a^{13} - \frac{609899}{1955907} a^{12} - \frac{2481448}{5867721} a^{11} + \frac{382}{2187} a^{10} - \frac{347}{729} a^{9} + \frac{689}{2187} a^{8} - \frac{689}{2187} a^{7} + \frac{40}{729} a^{6} + \frac{76612}{217323} a^{5} - \frac{76612}{217323} a^{4} - \frac{4171}{72441} a^{3} + \frac{253}{8049} a^{2} - \frac{253}{8049} a - \frac{253}{2683}$, $\frac{1}{17603163} a^{28} - \frac{1}{17603163} a^{27} - \frac{1}{5867721} a^{26} + \frac{10}{17603163} a^{25} - \frac{10}{17603163} a^{24} - \frac{10}{5867721} a^{23} + \frac{127}{17603163} a^{22} + \frac{692}{6561} a^{21} + \frac{692}{2187} a^{20} - \frac{332}{6561} a^{19} + \frac{332}{6561} a^{18} - \frac{887279}{17603163} a^{17} - \frac{889232}{5867721} a^{16} - \frac{697279}{17603163} a^{15} + \frac{689422}{17603163} a^{14} + \frac{694039}{5867721} a^{13} - \frac{525541}{17603163} a^{12} + \frac{540850}{17603163} a^{11} - \frac{347}{2187} a^{10} - \frac{769}{6561} a^{9} + \frac{769}{6561} a^{8} + \frac{769}{2187} a^{7} - \frac{285593}{651969} a^{6} + \frac{285593}{651969} a^{5} + \frac{68270}{217323} a^{4} + \frac{10985}{24147} a^{3} - \frac{10985}{24147} a^{2} - \frac{2936}{8049} a - \frac{746}{2683}$, $\frac{1}{52809489} a^{29} - \frac{1}{52809489} a^{28} - \frac{1}{17603163} a^{27} + \frac{10}{52809489} a^{26} - \frac{10}{52809489} a^{25} - \frac{10}{17603163} a^{24} + \frac{127}{52809489} a^{23} - \frac{127}{52809489} a^{22} + \frac{692}{6561} a^{21} + \frac{6229}{19683} a^{20} - \frac{6229}{19683} a^{19} + \frac{16715884}{52809489} a^{18} - \frac{889232}{17603163} a^{17} - \frac{697279}{52809489} a^{16} + \frac{689422}{52809489} a^{15} + \frac{694039}{17603163} a^{14} + \frac{17077622}{52809489} a^{13} - \frac{17062313}{52809489} a^{12} + \frac{534289}{17603163} a^{11} - \frac{7330}{19683} a^{10} + \frac{7330}{19683} a^{9} + \frac{769}{6561} a^{8} - \frac{937562}{1955907} a^{7} + \frac{937562}{1955907} a^{6} + \frac{285593}{651969} a^{5} + \frac{10985}{72441} a^{4} - \frac{10985}{72441} a^{3} - \frac{10985}{24147} a^{2} - \frac{1143}{2683} a + \frac{1143}{2683}$, $\frac{1}{633713868} a^{30} - \frac{1}{158428467} a^{29} + \frac{19}{633713868} a^{27} - \frac{10}{158428467} a^{26} + \frac{217}{633713868} a^{24} - \frac{127}{158428467} a^{23} + \frac{1}{236196} a^{21} - \frac{6229}{59049} a^{20} + \frac{56320393}{158428467} a^{19} - \frac{217}{23470884} a^{18} + \frac{15443}{59049} a^{17} - \frac{38911802}{158428467} a^{16} + \frac{19}{869292} a^{15} - \frac{13753}{59049} a^{14} + \frac{22533322}{158428467} a^{13} - \frac{1}{32196} a^{12} - \frac{15814}{59049} a^{11} - \frac{12353}{59049} a^{10} + \frac{1}{4} a^{9} - \frac{5619769}{17603163} a^{8} + \frac{937562}{5867721} a^{7} - \frac{1}{4} a^{6} + \frac{262825}{651969} a^{5} - \frac{10985}{217323} a^{4} + \frac{1}{4} a^{3} + \frac{9938}{24147} a^{2} - \frac{1540}{8049} a - \frac{1}{4}$, $\frac{1}{44224791353675316} a^{31} + \frac{28594415}{44224791353675316} a^{30} - \frac{14013983}{3685399279472943} a^{29} + \frac{53790139}{44224791353675316} a^{28} + \frac{704664281}{44224791353675316} a^{27} - \frac{140139830}{3685399279472943} a^{26} + \frac{537901417}{44224791353675316} a^{25} + \frac{7818692015}{44224791353675316} a^{24} - \frac{1779775841}{3685399279472943} a^{23} + \frac{6831347923}{44224791353675316} a^{22} - \frac{821161166209}{16483336322652} a^{21} - \frac{5125484511955748}{11056197838418829} a^{20} + \frac{1865681423187359}{14741597117891772} a^{19} - \frac{4295353905868117}{44224791353675316} a^{18} - \frac{3886316820431282}{11056197838418829} a^{17} + \frac{4303767125304725}{14741597117891772} a^{16} - \frac{13989692392676707}{44224791353675316} a^{15} - \frac{352084612416752}{11056197838418829} a^{14} + \frac{4961486971755311}{14741597117891772} a^{13} + \frac{9477268736845667}{44224791353675316} a^{12} + \frac{2110578108376156}{11056197838418829} a^{11} + \frac{1966534567799}{5494445440884} a^{10} - \frac{12877790160661}{1637955235321308} a^{9} - \frac{157773269818457}{409488808830327} a^{8} + \frac{152320795521853}{545985078440436} a^{7} + \frac{1054621677247}{60665008715604} a^{6} - \frac{3312898030378}{15166252178901} a^{5} + \frac{9170342938445}{20221669571868} a^{4} - \frac{86354569933}{2246852174652} a^{3} + \frac{231815859388}{561713043663} a^{2} + \frac{50525557181}{748950724884} a + \frac{21230664213}{249650241628}$, $\frac{1}{132674374061025948} a^{32} - \frac{1}{132674374061025948} a^{31} + \frac{29238911}{44224791353675316} a^{30} + \frac{1044866827}{132674374061025948} a^{29} - \frac{1395733771}{132674374061025948} a^{28} - \frac{840194443}{44224791353675316} a^{27} + \frac{10448668297}{132674374061025948} a^{26} - \frac{13957337737}{132674374061025948} a^{25} - \frac{7612493833}{44224791353675316} a^{24} + \frac{132698087299}{132674374061025948} a^{23} - \frac{177258189187}{132674374061025948} a^{22} + \frac{9453741043717543}{132674374061025948} a^{21} - \frac{14755078230935477}{44224791353675316} a^{20} + \frac{18812242567240439}{132674374061025948} a^{19} - \frac{60734566153240835}{132674374061025948} a^{18} + \frac{10984990178151265}{44224791353675316} a^{17} + \frac{1435263710359001}{132674374061025948} a^{16} + \frac{45928468831043155}{132674374061025948} a^{15} + \frac{21036329021981983}{44224791353675316} a^{14} - \frac{8414309825021929}{132674374061025948} a^{13} + \frac{13520768722162537}{132674374061025948} a^{12} + \frac{20485320139475509}{44224791353675316} a^{11} - \frac{629368259146375}{1637955235321308} a^{10} + \frac{334149918188345}{4913865705963924} a^{9} - \frac{805342369068079}{1637955235321308} a^{8} - \frac{13393036025843}{60665008715604} a^{7} - \frac{27365079161339}{181995026146812} a^{6} - \frac{3251408266847}{60665008715604} a^{5} + \frac{938759033261}{2246852174652} a^{4} + \frac{2240708391665}{6740556523956} a^{3} + \frac{424210010441}{2246852174652} a^{2} + \frac{162752885357}{748950724884} a + \frac{66020788121}{249650241628}$, $\frac{1}{398023122183077844} a^{33} - \frac{1}{398023122183077844} a^{32} - \frac{1}{132674374061025948} a^{31} - \frac{67970270}{99505780545769461} a^{30} + \frac{2419271837}{398023122183077844} a^{29} - \frac{443915839}{132674374061025948} a^{28} - \frac{2290245761}{99505780545769461} a^{27} + \frac{24192718343}{398023122183077844} a^{26} - \frac{4439158417}{132674374061025948} a^{25} - \frac{24737654900}{99505780545769461} a^{24} + \frac{307247523029}{398023122183077844} a^{23} - \frac{7566706855301}{398023122183077844} a^{22} + \frac{2341147151331721}{33168593515256487} a^{21} - \frac{18715634995470277}{398023122183077844} a^{20} + \frac{122214955983133945}{398023122183077844} a^{19} + \frac{14906536754615551}{33168593515256487} a^{18} - \frac{180682817952003403}{398023122183077844} a^{17} + \frac{254636639091283}{398023122183077844} a^{16} + \frac{13264779540573055}{33168593515256487} a^{15} + \frac{75988408700735555}{398023122183077844} a^{14} + \frac{118165200470906593}{398023122183077844} a^{13} + \frac{4426791536246635}{33168593515256487} a^{12} + \frac{1025313963460585}{4913865705963924} a^{11} + \frac{625913220433787}{4913865705963924} a^{10} + \frac{56034022321492}{136496269610109} a^{9} + \frac{203454476552419}{545985078440436} a^{8} + \frac{46143438988555}{181995026146812} a^{7} + \frac{1724231306756}{5055417392967} a^{6} + \frac{1375581142703}{20221669571868} a^{5} - \frac{171205428205}{6740556523956} a^{4} + \frac{277832431946}{561713043663} a^{3} - \frac{580897479011}{2246852174652} a^{2} - \frac{305834551243}{748950724884} a - \frac{32355177521}{249650241628}$, $\frac{1}{1194069366549233532} a^{34} - \frac{1}{1194069366549233532} a^{33} - \frac{1}{398023122183077844} a^{32} + \frac{5}{597034683274616766} a^{31} + \frac{844237637}{1194069366549233532} a^{30} + \frac{702866717}{398023122183077844} a^{29} - \frac{2742775321}{597034683274616766} a^{28} - \frac{416137141}{1194069366549233532} a^{27} + \frac{7028667143}{398023122183077844} a^{26} - \frac{27427753075}{597034683274616766} a^{25} + \frac{18633044789}{1194069366549233532} a^{24} - \frac{7129782701465}{1194069366549233532} a^{23} + \frac{1116818331833}{199011561091538922} a^{22} + \frac{130125341414196815}{1194069366549233532} a^{21} + \frac{155548449717181465}{1194069366549233532} a^{20} - \frac{55770324303110245}{199011561091538922} a^{19} + \frac{261147164912118221}{1194069366549233532} a^{18} + \frac{372857894786300719}{1194069366549233532} a^{17} - \frac{39840056616281767}{199011561091538922} a^{16} + \frac{154509034558328555}{1194069366549233532} a^{15} - \frac{430098475617727979}{1194069366549233532} a^{14} + \frac{85916288568594935}{199011561091538922} a^{13} + \frac{2256530262556675}{44224791353675316} a^{12} - \frac{1019130163574647}{44224791353675316} a^{11} - \frac{3428294587179251}{7370798558945886} a^{10} - \frac{300183611489561}{1637955235321308} a^{9} + \frac{140599207547341}{1637955235321308} a^{8} - \frac{71468685694219}{272992539220218} a^{7} - \frac{27013604348869}{60665008715604} a^{6} + \frac{24229636820921}{60665008715604} a^{5} - \frac{1781235400673}{10110834785934} a^{4} - \frac{2703841837535}{6740556523956} a^{3} + \frac{700204572701}{2246852174652} a^{2} + \frac{305758738313}{748950724884} a - \frac{17017037980}{62412560407}$, $\frac{1}{3582208099647700596} a^{35} - \frac{1}{3582208099647700596} a^{34} - \frac{1}{1194069366549233532} a^{33} + \frac{5}{1791104049823850298} a^{32} - \frac{5}{1791104049823850298} a^{31} - \frac{31582753}{298517341637308383} a^{30} + \frac{1455853037}{1791104049823850298} a^{29} - \frac{697867025}{1791104049823850298} a^{28} - \frac{949005772}{298517341637308383} a^{27} + \frac{14558530505}{1791104049823850298} a^{26} - \frac{6978670385}{1791104049823850298} a^{25} - \frac{1880422106111}{895552024911925149} a^{24} + \frac{1294560265655}{597034683274616766} a^{23} + \frac{11007733266223}{1791104049823850298} a^{22} - \frac{4914431978356286}{895552024911925149} a^{21} + \frac{1051526817758171}{597034683274616766} a^{20} - \frac{406826670876371489}{1791104049823850298} a^{19} + \frac{182637771513819166}{895552024911925149} a^{18} - \frac{143716464125255329}{597034683274616766} a^{17} - \frac{487719966619522127}{1791104049823850298} a^{16} - \frac{97415998101278360}{895552024911925149} a^{15} - \frac{214704050623849141}{597034683274616766} a^{14} - \frac{15812739716874883}{66337187030512974} a^{13} - \frac{6593331638621494}{33168593515256487} a^{12} + \frac{1158805657932913}{22112395676837658} a^{11} - \frac{3018895546890235}{7370798558945886} a^{10} - \frac{3191652703546}{15166252178901} a^{9} + \frac{386618897325247}{818977617660654} a^{8} - \frac{122371332986525}{272992539220218} a^{7} - \frac{649113203183}{1685139130989} a^{6} + \frac{14816178895595}{30332504357802} a^{5} + \frac{4639694472889}{10110834785934} a^{4} + \frac{785301167422}{1685139130989} a^{3} - \frac{651342412183}{2246852174652} a^{2} + \frac{304335814685}{748950724884} a + \frac{5320253099}{249650241628}$, $\frac{1}{10746624298943101788} a^{36} - \frac{1}{10746624298943101788} a^{35} - \frac{1}{3582208099647700596} a^{34} + \frac{5}{5373312149471550894} a^{33} - \frac{5}{5373312149471550894} a^{32} - \frac{5}{1791104049823850298} a^{31} + \frac{1204537849}{2686656074735775447} a^{30} - \frac{36893895089}{5373312149471550894} a^{29} + \frac{9085864099}{1791104049823850298} a^{28} + \frac{63772607644}{2686656074735775447} a^{27} - \frac{368938951025}{5373312149471550894} a^{26} - \frac{3426211537351}{5373312149471550894} a^{25} + \frac{839880776107}{895552024911925149} a^{24} + \frac{6410837702095}{5373312149471550894} a^{23} - \frac{33526160381491}{5373312149471550894} a^{22} - \frac{19128180709369106}{895552024911925149} a^{21} - \frac{2040055422956649461}{5373312149471550894} a^{20} - \frac{2351386709354214529}{5373312149471550894} a^{19} - \frac{155496310641135386}{895552024911925149} a^{18} - \frac{122031958771897121}{5373312149471550894} a^{17} - \frac{959654355708304735}{5373312149471550894} a^{16} - \frac{280319935740469424}{895552024911925149} a^{15} - \frac{54183374219339647}{199011561091538922} a^{14} + \frac{38394147186816505}{199011561091538922} a^{13} - \frac{14446490736124522}{33168593515256487} a^{12} - \frac{2475222877432475}{7370798558945886} a^{11} + \frac{3489300468638675}{7370798558945886} a^{10} + \frac{489272950510900}{1228466426490981} a^{9} - \frac{58907080786085}{272992539220218} a^{8} + \frac{75160167566635}{272992539220218} a^{7} - \frac{21744159197914}{45498756536703} a^{6} + \frac{4391093071331}{30332504357802} a^{5} - \frac{4702906785869}{10110834785934} a^{4} + \frac{6585793915}{6740556523956} a^{3} - \frac{909929684839}{2246852174652} a^{2} - \frac{74201510653}{249650241628} a + \frac{13267702943}{124825120814}$, $\frac{1}{32239872896829305364} a^{37} - \frac{1}{32239872896829305364} a^{36} - \frac{1}{10746624298943101788} a^{35} + \frac{5}{16119936448414652682} a^{34} - \frac{5}{16119936448414652682} a^{33} - \frac{5}{5373312149471550894} a^{32} + \frac{127}{32239872896829305364} a^{31} + \frac{12982906673}{32239872896829305364} a^{30} + \frac{245257771}{5373312149471550894} a^{29} - \frac{53403172667}{32239872896829305364} a^{28} + \frac{86465705039}{32239872896829305364} a^{27} - \frac{3691429727191}{16119936448414652682} a^{26} + \frac{2287847732197}{10746624298943101788} a^{25} + \frac{23407920290057}{32239872896829305364} a^{24} - \frac{36894431392459}{16119936448414652682} a^{23} + \frac{22397848767967}{10746624298943101788} a^{22} - \frac{1942407188529214837}{32239872896829305364} a^{21} - \frac{5794353088660575421}{16119936448414652682} a^{20} + \frac{5188303527046596805}{10746624298943101788} a^{19} - \frac{7853126162628034075}{32239872896829305364} a^{18} - \frac{3605612101736517925}{16119936448414652682} a^{17} - \frac{1274140395893643737}{10746624298943101788} a^{16} + \frac{405235073611586533}{1194069366549233532} a^{15} - \frac{205313799578046725}{597034683274616766} a^{14} - \frac{141784708659896281}{398023122183077844} a^{13} - \frac{8095086180165487}{44224791353675316} a^{12} - \frac{10571775853534603}{22112395676837658} a^{11} + \frac{885404075758255}{14741597117891772} a^{10} + \frac{1393984536721795}{4913865705963924} a^{9} - \frac{136483524497893}{272992539220218} a^{8} - \frac{236312237816833}{545985078440436} a^{7} - \frac{15583109022239}{60665008715604} a^{6} - \frac{1967699686711}{30332504357802} a^{5} + \frac{3056656890755}{10110834785934} a^{4} + \frac{92028532453}{1123426087326} a^{3} - \frac{390834509863}{2246852174652} a^{2} + \frac{84147236359}{249650241628} a - \frac{29047258363}{249650241628}$, $\frac{1}{96719618690487916092} a^{38} - \frac{1}{96719618690487916092} a^{37} - \frac{1}{32239872896829305364} a^{36} + \frac{5}{48359809345243958046} a^{35} - \frac{5}{48359809345243958046} a^{34} - \frac{5}{16119936448414652682} a^{33} + \frac{127}{96719618690487916092} a^{32} - \frac{127}{96719618690487916092} a^{31} - \frac{589525018}{8059968224207326341} a^{30} - \frac{869717885039}{96719618690487916092} a^{29} + \frac{898015084379}{96719618690487916092} a^{28} - \frac{1209485340890}{24179904672621979023} a^{27} - \frac{433201309043}{32239872896829305364} a^{26} + \frac{31172875599857}{96719618690487916092} a^{25} - \frac{12142604935358}{24179904672621979023} a^{24} - \frac{12159474055781}{32239872896829305364} a^{23} + \frac{335975163276803}{96719618690487916092} a^{22} - \frac{1904745902558866928}{24179904672621979023} a^{21} - \frac{7133841034902202487}{32239872896829305364} a^{20} - \frac{25270871533596945103}{96719618690487916092} a^{19} - \frac{6731405161587188711}{24179904672621979023} a^{18} + \frac{11885257089416002063}{32239872896829305364} a^{17} - \frac{884192388974990231}{3582208099647700596} a^{16} + \frac{258436030487582897}{895552024911925149} a^{15} + \frac{370384093537915307}{1194069366549233532} a^{14} + \frac{63306873078894761}{132674374061025948} a^{13} + \frac{12248713624009852}{33168593515256487} a^{12} + \frac{2451713320559971}{44224791353675316} a^{11} - \frac{1487166489381425}{14741597117891772} a^{10} + \frac{568730880144643}{1228466426490981} a^{9} - \frac{190607819908715}{545985078440436} a^{8} + \frac{2653704538427}{60665008715604} a^{7} - \frac{5315621434798}{15166252178901} a^{6} - \frac{3729260979941}{10110834785934} a^{5} + \frac{49433310785}{1123426087326} a^{4} - \frac{31034567675}{2246852174652} a^{3} - \frac{85605002713}{748950724884} a^{2} - \frac{283777865059}{748950724884} a + \frac{49760573921}{124825120814}$, $\frac{1}{290158856071463748276} a^{39} - \frac{1}{290158856071463748276} a^{38} - \frac{1}{96719618690487916092} a^{37} + \frac{5}{145079428035731874138} a^{36} - \frac{5}{145079428035731874138} a^{35} - \frac{5}{48359809345243958046} a^{34} + \frac{127}{290158856071463748276} a^{33} - \frac{127}{290158856071463748276} a^{32} - \frac{127}{96719618690487916092} a^{31} + \frac{44126511082}{72539714017865937069} a^{30} - \frac{2029210757461}{290158856071463748276} a^{29} - \frac{6074388339683}{290158856071463748276} a^{28} + \frac{1226729125495}{24179904672621979023} a^{27} + \frac{1900617181457}{290158856071463748276} a^{26} - \frac{60743883407117}{290158856071463748276} a^{25} + \frac{12664429854688}{24179904672621979023} a^{24} - \frac{35782518636877}{290158856071463748276} a^{23} - \frac{771447319242611}{290158856071463748276} a^{22} + \frac{3076345923863542093}{24179904672621979023} a^{21} - \frac{85472619559266729235}{290158856071463748276} a^{20} + \frac{438730288032367327}{290158856071463748276} a^{19} + \frac{9355740599199542032}{24179904672621979023} a^{18} - \frac{5246336164249028147}{10746624298943101788} a^{17} - \frac{4339719169343855149}{10746624298943101788} a^{16} - \frac{444446333320319929}{895552024911925149} a^{15} + \frac{65560543870493057}{398023122183077844} a^{14} + \frac{1038362606156407}{398023122183077844} a^{13} + \frac{16056190423582111}{33168593515256487} a^{12} + \frac{2290700501886871}{4913865705963924} a^{11} - \frac{107632952496419}{545985078440436} a^{10} - \frac{140877745336174}{1228466426490981} a^{9} - \frac{198334185757103}{545985078440436} a^{8} + \frac{266974510112491}{545985078440436} a^{7} - \frac{8957514151145}{181995026146812} a^{6} + \frac{360242276995}{10110834785934} a^{5} - \frac{1036242651755}{10110834785934} a^{4} + \frac{1386925587613}{3370278261978} a^{3} - \frac{112769309511}{249650241628} a^{2} + \frac{132948970925}{748950724884} a - \frac{53358221139}{249650241628}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{59541067}{132674374061025948} a^{34} + \frac{6160}{1425856203} a^{23} + \frac{59541067}{1425856203} a^{12} - \frac{573797262679}{748950724884} a \) (order $66$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{429}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{13}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{33}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-143})\), \(\Q(\sqrt{-11}, \sqrt{-39})\), \(\Q(\zeta_{11})^+\), 8.0.33871089681.2, \(\Q(\zeta_{33})^+\), 10.10.79589952003133.1, 10.10.212743941704374509.1, 10.0.52089208083.1, \(\Q(\zeta_{11})\), 10.0.19340358336761319.3, 10.0.875489472034463.1, 20.20.45259984731914299465343386928991081.1, \(\Q(\zeta_{33})\), 20.0.45259984731914299465343386928991081.3, 20.0.374049460594333053432589974619761.1, 20.0.766481815643182771348259698369.1, 20.0.45259984731914299465343386928991081.4, 20.0.45259984731914299465343386928991081.5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
13Data not computed