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Properties

Label 36.0.49997585682...0625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 5^{18}\cdot 7^{30}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number $95904$ (GRH)
Class group $[2, 2, 2, 18, 666]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31530241, -19886784, -2375280, 4778248, -7662084, 12034116, -5874034, 5792076, -768960, 3953440, -642219, 3617676, -837713, 1184688, 2297349, -1959888, 5371470, -3268836, 5624552, -2551176, 3855771, -1274184, 1912974, -438696, 708976, -106056, 196911, -17788, 40425, -1980, 5951, -132, 594, -4, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241)
 
gp: K = bnfinit(x^36 + 36*x^34 - 4*x^33 + 594*x^32 - 132*x^31 + 5951*x^30 - 1980*x^29 + 40425*x^28 - 17788*x^27 + 196911*x^26 - 106056*x^25 + 708976*x^24 - 438696*x^23 + 1912974*x^22 - 1274184*x^21 + 3855771*x^20 - 2551176*x^19 + 5624552*x^18 - 3268836*x^17 + 5371470*x^16 - 1959888*x^15 + 2297349*x^14 + 1184688*x^13 - 837713*x^12 + 3617676*x^11 - 642219*x^10 + 3953440*x^9 - 768960*x^8 + 5792076*x^7 - 5874034*x^6 + 12034116*x^5 - 7662084*x^4 + 4778248*x^3 - 2375280*x^2 - 19886784*x + 31530241, 1)
 

Normalized defining polynomial

\( x^{36} + 36 x^{34} - 4 x^{33} + 594 x^{32} - 132 x^{31} + 5951 x^{30} - 1980 x^{29} + 40425 x^{28} - 17788 x^{27} + 196911 x^{26} - 106056 x^{25} + 708976 x^{24} - 438696 x^{23} + 1912974 x^{22} - 1274184 x^{21} + 3855771 x^{20} - 2551176 x^{19} + 5624552 x^{18} - 3268836 x^{17} + 5371470 x^{16} - 1959888 x^{15} + 2297349 x^{14} + 1184688 x^{13} - 837713 x^{12} + 3617676 x^{11} - 642219 x^{10} + 3953440 x^{9} - 768960 x^{8} + 5792076 x^{7} - 5874034 x^{6} + 12034116 x^{5} - 7662084 x^{4} + 4778248 x^{3} - 2375280 x^{2} - 19886784 x + 31530241 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4999758568289868528789868885747458073284974388119052886962890625=3^{54}\cdot 5^{18}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(131,·)$, $\chi_{315}(134,·)$, $\chi_{315}(139,·)$, $\chi_{315}(16,·)$, $\chi_{315}(146,·)$, $\chi_{315}(19,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(26,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(34,·)$, $\chi_{315}(41,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(304,·)$, $\chi_{315}(179,·)$, $\chi_{315}(311,·)$, $\chi_{315}(199,·)$, $\chi_{315}(74,·)$, $\chi_{315}(206,·)$, $\chi_{315}(211,·)$, $\chi_{315}(94,·)$, $\chi_{315}(101,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(106,·)$, $\chi_{315}(236,·)$, $\chi_{315}(239,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(251,·)$, $\chi_{315}(124,·)$, $\chi_{315}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{10946} a^{21} + \frac{21}{10946} a^{19} + \frac{189}{10946} a^{17} + \frac{476}{5473} a^{15} + \frac{1470}{5473} a^{13} - \frac{401}{842} a^{11} - \frac{303}{842} a^{9} + \frac{198}{421} a^{7} + \frac{2079}{10946} a^{5} + \frac{385}{10946} a^{3} + \frac{21}{10946} a - \frac{4181}{10946}$, $\frac{1}{10946} a^{22} + \frac{21}{10946} a^{20} + \frac{189}{10946} a^{18} + \frac{476}{5473} a^{16} + \frac{1470}{5473} a^{14} - \frac{401}{842} a^{12} - \frac{303}{842} a^{10} + \frac{198}{421} a^{8} + \frac{2079}{10946} a^{6} + \frac{385}{10946} a^{4} + \frac{21}{10946} a^{2} - \frac{4181}{10946} a$, $\frac{1}{10946} a^{23} - \frac{126}{5473} a^{19} - \frac{3017}{10946} a^{17} + \frac{2420}{5473} a^{15} - \frac{1277}{10946} a^{13} - \frac{151}{421} a^{11} + \frac{23}{842} a^{9} + \frac{3431}{10946} a^{7} + \frac{255}{5473} a^{5} + \frac{1441}{5473} a^{3} - \frac{4181}{10946} a^{2} - \frac{441}{10946} a + \frac{233}{10946}$, $\frac{1}{10946} a^{24} - \frac{126}{5473} a^{20} + \frac{1228}{5473} a^{18} + \frac{2420}{5473} a^{16} - \frac{1}{2} a^{15} + \frac{2098}{5473} a^{14} - \frac{1}{2} a^{13} - \frac{151}{421} a^{12} - \frac{199}{421} a^{10} + \frac{3431}{10946} a^{8} - \frac{1}{2} a^{7} - \frac{4963}{10946} a^{6} - \frac{1}{2} a^{5} + \frac{1441}{5473} a^{4} - \frac{4181}{10946} a^{3} - \frac{441}{10946} a^{2} + \frac{233}{10946} a - \frac{1}{2}$, $\frac{1}{10946} a^{25} + \frac{175}{842} a^{19} - \frac{87}{421} a^{17} - \frac{2185}{10946} a^{15} + \frac{1786}{5473} a^{13} + \frac{11}{842} a^{11} - \frac{4057}{10946} a^{9} - \frac{2384}{5473} a^{7} + \frac{691}{5473} a^{5} - \frac{4181}{10946} a^{4} - \frac{1935}{10946} a^{3} + \frac{233}{10946} a^{2} + \frac{2646}{5473} a - \frac{1398}{5473}$, $\frac{1}{10946} a^{26} + \frac{175}{842} a^{20} - \frac{87}{421} a^{18} - \frac{2185}{10946} a^{16} + \frac{1786}{5473} a^{14} + \frac{11}{842} a^{12} - \frac{4057}{10946} a^{10} - \frac{2384}{5473} a^{8} + \frac{691}{5473} a^{6} - \frac{4181}{10946} a^{5} - \frac{1935}{10946} a^{4} + \frac{233}{10946} a^{3} + \frac{2646}{5473} a^{2} - \frac{1398}{5473} a$, $\frac{1}{186082} a^{27} - \frac{7}{186082} a^{25} + \frac{1}{186082} a^{23} - \frac{1}{186082} a^{21} - \frac{2569}{14314} a^{19} - \frac{2}{17} a^{18} - \frac{45576}{93041} a^{17} - \frac{2}{17} a^{16} + \frac{2715}{10946} a^{15} + \frac{2}{17} a^{14} + \frac{418}{5473} a^{13} - \frac{4}{17} a^{12} - \frac{928}{5473} a^{11} - \frac{7}{17} a^{10} - \frac{4259}{93041} a^{9} + \frac{6}{17} a^{8} + \frac{11669}{186082} a^{7} + \frac{83387}{186082} a^{6} - \frac{1481}{10946} a^{5} + \frac{25696}{93041} a^{4} - \frac{66429}{186082} a^{3} + \frac{23061}{93041} a^{2} + \frac{46071}{186082} a + \frac{34633}{186082}$, $\frac{1}{186082} a^{28} - \frac{7}{186082} a^{26} + \frac{1}{186082} a^{24} - \frac{1}{186082} a^{22} - \frac{2569}{14314} a^{20} - \frac{2}{17} a^{19} + \frac{1889}{186082} a^{18} - \frac{2}{17} a^{17} + \frac{2715}{10946} a^{16} - \frac{13}{34} a^{15} - \frac{4637}{10946} a^{14} + \frac{9}{34} a^{13} - \frac{928}{5473} a^{12} - \frac{7}{17} a^{11} + \frac{84523}{186082} a^{10} + \frac{6}{17} a^{9} + \frac{11669}{186082} a^{8} - \frac{4827}{93041} a^{7} + \frac{1996}{5473} a^{6} - \frac{41649}{186082} a^{5} - \frac{66429}{186082} a^{4} + \frac{23061}{93041} a^{3} + \frac{46071}{186082} a^{2} + \frac{34633}{186082} a - \frac{1}{2}$, $\frac{1}{6557470319842} a^{29} + \frac{12238}{3278735159921} a^{28} + \frac{29}{6557470319842} a^{27} + \frac{71053826}{3278735159921} a^{26} - \frac{103305}{6557470319842} a^{25} + \frac{45549781}{3278735159921} a^{24} + \frac{30796113}{6557470319842} a^{23} - \frac{185995755}{6557470319842} a^{22} + \frac{70144298}{3278735159921} a^{21} - \frac{478134856475}{6557470319842} a^{20} - \frac{93600027659}{3278735159921} a^{19} - \frac{808825282751}{3278735159921} a^{18} + \frac{116838667338}{252210396917} a^{17} + \frac{133279503293}{504420793834} a^{16} + \frac{796955029327}{3278735159921} a^{15} + \frac{29853957775}{192866774113} a^{14} - \frac{3000699778711}{6557470319842} a^{13} + \frac{396644725233}{6557470319842} a^{12} - \frac{88890202373}{252210396917} a^{11} + \frac{1574529199077}{3278735159921} a^{10} + \frac{1350791062839}{3278735159921} a^{9} - \frac{1494789215638}{3278735159921} a^{8} - \frac{1619857863435}{6557470319842} a^{7} + \frac{779742212257}{6557470319842} a^{6} - \frac{1825125509731}{6557470319842} a^{5} - \frac{2102702539823}{6557470319842} a^{4} - \frac{1636459959438}{3278735159921} a^{3} - \frac{1444761878466}{3278735159921} a^{2} + \frac{2135469084367}{6557470319842} a + \frac{387135740986}{3278735159921}$, $\frac{1}{6557470319842} a^{30} + \frac{15}{3278735159921} a^{28} + \frac{219562}{3278735159921} a^{27} + \frac{405}{6557470319842} a^{26} + \frac{5928174}{3278735159921} a^{25} + \frac{33492195}{6557470319842} a^{24} + \frac{71138088}{3278735159921} a^{23} + \frac{204677353}{6557470319842} a^{22} + \frac{19954951}{504420793834} a^{21} - \frac{4141288221}{6557470319842} a^{20} - \frac{10985196705}{6557470319842} a^{19} - \frac{31161850173}{3278735159921} a^{18} - \frac{9673612893}{504420793834} a^{17} - \frac{14448054238}{252210396917} a^{16} + \frac{1015846644265}{6557470319842} a^{15} - \frac{634829568900}{3278735159921} a^{14} - \frac{231449037625}{504420793834} a^{13} - \frac{1405149745175}{3278735159921} a^{12} - \frac{2907333273257}{6557470319842} a^{11} + \frac{832914510039}{6557470319842} a^{10} + \frac{712810668212}{3278735159921} a^{9} - \frac{323101112917}{6557470319842} a^{8} - \frac{1553656964778}{3278735159921} a^{7} + \frac{1322754874990}{3278735159921} a^{6} - \frac{2909769944891}{6557470319842} a^{5} - \frac{329700782287}{3278735159921} a^{4} + \frac{526443664912}{3278735159921} a^{3} + \frac{62143740323}{385733548226} a^{2} - \frac{122533724124}{252210396917} a - \frac{37161884305}{252210396917}$, $\frac{1}{6557470319842} a^{31} - \frac{147578}{3278735159921} a^{28} - \frac{465}{6557470319842} a^{27} - \frac{57851173}{6557470319842} a^{26} + \frac{36591345}{6557470319842} a^{25} - \frac{97206188}{3278735159921} a^{24} - \frac{60065730}{3278735159921} a^{23} - \frac{151458757}{6557470319842} a^{22} + \frac{37097977}{6557470319842} a^{21} + \frac{192504182037}{6557470319842} a^{20} - \frac{489315614207}{3278735159921} a^{19} + \frac{523756880089}{3278735159921} a^{18} - \frac{1603108642014}{3278735159921} a^{17} + \frac{1507386261117}{6557470319842} a^{16} + \frac{570923968284}{3278735159921} a^{15} - \frac{1543696220151}{3278735159921} a^{14} + \frac{85450733767}{192866774113} a^{13} - \frac{2221316316631}{6557470319842} a^{12} - \frac{2136236486305}{6557470319842} a^{11} + \frac{1390320289785}{3278735159921} a^{10} + \frac{40594029943}{504420793834} a^{9} + \frac{1343109118587}{6557470319842} a^{8} - \frac{1888881357369}{6557470319842} a^{7} - \frac{1115310367082}{3278735159921} a^{6} - \frac{717528481964}{3278735159921} a^{5} + \frac{1601960102677}{6557470319842} a^{4} - \frac{1602150667788}{3278735159921} a^{3} + \frac{10701780357}{192866774113} a^{2} + \frac{71880883651}{385733548226} a - \frac{1208146827090}{3278735159921}$, $\frac{1}{6557470319842} a^{32} + \frac{51593}{3278735159921} a^{28} - \frac{7025984}{3278735159921} a^{27} + \frac{2894169}{6557470319842} a^{26} + \frac{219697281}{6557470319842} a^{25} + \frac{81355520}{3278735159921} a^{24} - \frac{217247269}{6557470319842} a^{23} - \frac{293712573}{6557470319842} a^{22} - \frac{124421774}{3278735159921} a^{21} + \frac{139787780345}{6557470319842} a^{20} + \frac{16873788735}{6557470319842} a^{19} - \frac{60031627725}{3278735159921} a^{18} - \frac{710141652439}{3278735159921} a^{17} + \frac{921921718513}{3278735159921} a^{16} - \frac{1393886782843}{6557470319842} a^{15} + \frac{203266366027}{504420793834} a^{14} - \frac{1131768051116}{3278735159921} a^{13} + \frac{2575267333209}{6557470319842} a^{12} - \frac{28542334011}{252210396917} a^{11} - \frac{283214456467}{3278735159921} a^{10} - \frac{1511349968328}{3278735159921} a^{9} + \frac{1929276729561}{6557470319842} a^{8} + \frac{1088048171747}{6557470319842} a^{7} + \frac{972677600603}{6557470319842} a^{6} - \frac{2907147115055}{6557470319842} a^{5} + \frac{3153895105367}{6557470319842} a^{4} + \frac{195484891}{1198149154} a^{3} + \frac{1504802598831}{6557470319842} a^{2} - \frac{71950513075}{385733548226} a + \frac{356705315256}{3278735159921}$, $\frac{1}{6557470319842} a^{33} - \frac{1187736}{3278735159921} a^{28} - \frac{98225}{6557470319842} a^{27} - \frac{139697253}{6557470319842} a^{26} + \frac{19499192}{3278735159921} a^{25} - \frac{98252685}{3278735159921} a^{24} + \frac{34601197}{3278735159921} a^{23} - \frac{92487807}{3278735159921} a^{22} - \frac{176658901}{6557470319842} a^{21} + \frac{89260645158}{3278735159921} a^{20} - \frac{1570220861921}{6557470319842} a^{19} - \frac{744787523645}{3278735159921} a^{18} + \frac{687754268211}{3278735159921} a^{17} + \frac{1181177070001}{6557470319842} a^{16} - \frac{868874492288}{3278735159921} a^{15} + \frac{2236877975201}{6557470319842} a^{14} - \frac{3271426145159}{6557470319842} a^{13} - \frac{221784248575}{504420793834} a^{12} - \frac{173949192245}{385733548226} a^{11} - \frac{318249567853}{3278735159921} a^{10} + \frac{1782869432423}{6557470319842} a^{9} + \frac{326693637761}{3278735159921} a^{8} - \frac{2716141651135}{6557470319842} a^{7} + \frac{556609336635}{6557470319842} a^{6} - \frac{298584471688}{3278735159921} a^{5} + \frac{1523353189387}{6557470319842} a^{4} - \frac{193364395447}{3278735159921} a^{3} + \frac{816303705630}{3278735159921} a^{2} - \frac{952782060709}{6557470319842} a - \frac{272828370034}{3278735159921}$, $\frac{1}{6557470319842} a^{34} - \frac{3519203}{6557470319842} a^{28} - \frac{329203}{6557470319842} a^{27} - \frac{49240206}{3278735159921} a^{26} + \frac{131655283}{6557470319842} a^{25} + \frac{92090845}{3278735159921} a^{24} - \frac{5977991}{192866774113} a^{23} - \frac{164735271}{6557470319842} a^{22} + \frac{149834491}{6557470319842} a^{21} + \frac{333131900251}{6557470319842} a^{20} - \frac{323794103151}{6557470319842} a^{19} + \frac{250734694266}{3278735159921} a^{18} + \frac{458878117981}{3278735159921} a^{17} - \frac{2602471167481}{6557470319842} a^{16} + \frac{635135407997}{6557470319842} a^{15} + \frac{1568675489171}{3278735159921} a^{14} - \frac{2182562871395}{6557470319842} a^{13} - \frac{187337756661}{385733548226} a^{12} + \frac{1783520188583}{6557470319842} a^{11} - \frac{934669355859}{6557470319842} a^{10} + \frac{529240215956}{3278735159921} a^{9} + \frac{122856819591}{252210396917} a^{8} + \frac{919744795172}{3278735159921} a^{7} + \frac{3022825680787}{6557470319842} a^{6} + \frac{351355817751}{3278735159921} a^{5} - \frac{95268052803}{504420793834} a^{4} + \frac{1292009126149}{3278735159921} a^{3} - \frac{596441757126}{3278735159921} a^{2} + \frac{13394954433}{192866774113} a - \frac{257043476305}{6557470319842}$, $\frac{1}{6557470319842} a^{35} + \frac{582533}{385733548226} a^{28} + \frac{3576475}{6557470319842} a^{27} - \frac{240395721}{6557470319842} a^{26} + \frac{135592007}{3278735159921} a^{25} + \frac{34341681}{6557470319842} a^{24} + \frac{112473376}{3278735159921} a^{23} - \frac{139307797}{6557470319842} a^{22} + \frac{72809190}{3278735159921} a^{21} - \frac{592091457184}{3278735159921} a^{20} + \frac{105126979397}{504420793834} a^{19} + \frac{19055207393}{192866774113} a^{18} + \frac{2810006014441}{6557470319842} a^{17} + \frac{1199835981784}{3278735159921} a^{16} + \frac{123837091023}{385733548226} a^{15} - \frac{2616871093929}{6557470319842} a^{14} - \frac{1112018722739}{3278735159921} a^{13} - \frac{9705183267}{252210396917} a^{12} - \frac{456346362494}{3278735159921} a^{11} - \frac{669535128852}{3278735159921} a^{10} - \frac{2294137359039}{6557470319842} a^{9} - \frac{3086772664205}{6557470319842} a^{8} + \frac{126678951870}{3278735159921} a^{7} - \frac{2650357741815}{6557470319842} a^{6} - \frac{2173859726253}{6557470319842} a^{5} + \frac{2670018880951}{6557470319842} a^{4} + \frac{3213598992477}{6557470319842} a^{3} + \frac{1415480584970}{3278735159921} a^{2} + \frac{56110566346}{252210396917} a - \frac{1522679865033}{6557470319842}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{666}$, which has order $95904$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 816369751172.7767 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{-15}, \sqrt{21})\), 6.0.281302875.3, 6.0.13783840875.2, 6.0.2100875.1, 6.0.13783840875.1, 6.6.6751269.1, 6.0.2460375.1, 6.6.330812181.1, 6.0.5907360375.2, \(\Q(\zeta_{21})^+\), 6.0.8103375.1, 6.6.330812181.2, 6.0.5907360375.1, 9.9.62523502209.1, 12.0.712181767349390625.3, 12.0.1709948423405886890625.3, 12.0.3217569633140625.1, 12.0.1709948423405886890625.2, 18.0.2618850774742652270958169921875.4, \(\Q(\zeta_{63})^+\), 18.0.206148603259625688967552734375.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed