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Properties

Label 36.36.236...704.1
Degree $36$
Signature $[36, 0]$
Discriminant $2.361\times 10^{67}$
Root discriminant \(74.38\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_6^2$ (as 36T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144)
 
gp: K = bnfinit(y^36 - 72*y^34 + 2376*y^32 - 47624*y^30 + 647760*y^28 - 6327072*y^26 + 45809920*y^24 - 250210560*y^22 + 1039332096*y^20 - 3285680128*y^18 + 7858649088*y^16 - 14038161408*y^14 + 18332004352*y^12 - 16929153024*y^10 + 10498277376*y^8 - 4012965888*y^6 + 804519936*y^4 - 56623104*y^2 + 262144, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144)
 

\( x^{36} - 72 x^{34} + 2376 x^{32} - 47624 x^{30} + 647760 x^{28} - 6327072 x^{26} + 45809920 x^{24} + \cdots + 262144 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[36, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(23610692285332399309092778573219694177406932269512586359440570056704\) \(\medspace = 2^{54}\cdot 3^{54}\cdot 7^{30}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(74.38\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{3/2}7^{5/6}\approx 74.38326581667904$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(5,·)$, $\chi_{504}(257,·)$, $\chi_{504}(269,·)$, $\chi_{504}(17,·)$, $\chi_{504}(277,·)$, $\chi_{504}(25,·)$, $\chi_{504}(289,·)$, $\chi_{504}(421,·)$, $\chi_{504}(41,·)$, $\chi_{504}(173,·)$, $\chi_{504}(437,·)$, $\chi_{504}(185,·)$, $\chi_{504}(445,·)$, $\chi_{504}(169,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(205,·)$, $\chi_{504}(461,·)$, $\chi_{504}(337,·)$, $\chi_{504}(85,·)$, $\chi_{504}(377,·)$, $\chi_{504}(89,·)$, $\chi_{504}(37,·)$, $\chi_{504}(353,·)$, $\chi_{504}(101,·)$, $\chi_{504}(209,·)$, $\chi_{504}(361,·)$, $\chi_{504}(109,·)$, $\chi_{504}(125,·)$, $\chi_{504}(373,·)$, $\chi_{504}(425,·)$, $\chi_{504}(121,·)$, $\chi_{504}(293,·)$, $\chi_{504}(253,·)$, $\chi_{504}(341,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $35$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{32768}a^{30}-\frac{15}{8192}a^{28}+\frac{405}{8192}a^{26}-\frac{1625}{2048}a^{24}+\frac{8625}{1024}a^{22}-\frac{15939}{256}a^{20}+\frac{42061}{128}a^{18}-\frac{39969}{32}a^{16}+\frac{435915}{128}a^{14}-\frac{419353}{64}a^{12}+\frac{275835}{32}a^{10}-7344a^{8}+\frac{14721}{4}a^{6}-\frac{3555}{4}a^{4}+54a^{2}+1$, $\frac{1}{512}a^{18}-\frac{9}{128}a^{16}+\frac{135}{128}a^{14}-\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}-\frac{891}{8}a^{8}+\frac{693}{4}a^{6}-135a^{4}+\frac{81}{2}a^{2}-2$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-49a^{4}+\frac{49}{2}a^{2}-3$, $\frac{1}{16384}a^{28}-\frac{7}{2048}a^{26}+\frac{175}{2048}a^{24}-\frac{161}{128}a^{22}+\frac{12397}{1024}a^{20}-\frac{10241}{128}a^{18}+\frac{47481}{128}a^{16}-\frac{4845}{4}a^{14}+\frac{88179}{32}a^{12}-\frac{17017}{4}a^{10}+\frac{17017}{4}a^{8}-2548a^{6}+\frac{3185}{4}a^{4}-98a^{2}+2$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}-\frac{1519}{512}a^{18}+\frac{1449}{64}a^{16}-\frac{14553}{128}a^{14}+\frac{12103}{32}a^{12}-\frac{26169}{32}a^{10}+\frac{17523}{16}a^{8}-\frac{3311}{4}a^{6}+294a^{4}-\frac{63}{2}a^{2}+1$, $\frac{1}{32768}a^{30}-\frac{15}{8192}a^{28}+\frac{405}{8192}a^{26}-\frac{1625}{2048}a^{24}+\frac{8625}{1024}a^{22}-\frac{15939}{256}a^{20}+\frac{168245}{512}a^{18}-\frac{159885}{128}a^{16}+\frac{218025}{64}a^{14}-\frac{104975}{16}a^{12}+\frac{138567}{16}a^{10}-\frac{29835}{4}a^{8}+\frac{7735}{2}a^{6}-1050a^{4}+\frac{225}{2}a^{2}-3$, $\frac{1}{512}a^{18}-\frac{9}{128}a^{16}+\frac{135}{128}a^{14}-\frac{547}{64}a^{12}+\frac{1299}{32}a^{10}-\frac{459}{4}a^{8}+\frac{749}{4}a^{6}-\frac{645}{4}a^{4}+\frac{117}{2}a^{2}-4$, $\frac{1}{2}a^{2}-1$, $\frac{1}{65536}a^{32}-\frac{1}{1024}a^{30}+\frac{465}{16384}a^{28}-\frac{4059}{8192}a^{26}+\frac{5931}{1024}a^{24}-\frac{97957}{2048}a^{22}+\frac{293755}{1024}a^{20}-\frac{647615}{512}a^{18}+\frac{262599}{64}a^{16}-\frac{1241611}{128}a^{14}+\frac{32749}{2}a^{12}-\frac{152581}{8}a^{10}+\frac{232221}{16}a^{8}-\frac{26373}{4}a^{6}+\frac{6033}{4}a^{4}-\frac{225}{2}a^{2}$, $\frac{1}{65536}a^{32}-\frac{1}{1024}a^{30}+\frac{29}{1024}a^{28}-\frac{63}{128}a^{26}+\frac{2925}{512}a^{24}-\frac{1495}{32}a^{22}+\frac{8855}{32}a^{20}-\frac{4807}{4}a^{18}+\frac{245157}{64}a^{16}-\frac{17765}{2}a^{14}+\frac{29393}{2}a^{12}-\frac{537471}{32}a^{10}+\frac{100771}{8}a^{8}-\frac{45661}{8}a^{6}+\frac{2695}{2}a^{4}-\frac{231}{2}a^{2}$, $\frac{1}{16384}a^{28}-\frac{27}{8192}a^{26}+\frac{81}{1024}a^{24}-\frac{2277}{2048}a^{22}+\frac{5197}{512}a^{20}-\frac{32299}{512}a^{18}+\frac{34799}{128}a^{16}-\frac{103851}{128}a^{14}+\frac{105117}{64}a^{12}-\frac{4303}{2}a^{10}+\frac{13189}{8}a^{8}-\frac{2201}{4}a^{6}-63a^{4}+\frac{147}{2}a^{2}-6$, $\frac{1}{4}a^{4}-2a^{2}+3$, $\frac{1}{1024}a^{20}-\frac{5}{128}a^{18}+\frac{85}{128}a^{16}-\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}-\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}-330a^{6}+\frac{825}{4}a^{4}-50a^{2}+3$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}-\frac{12597}{128}a^{16}+\frac{12597}{32}a^{14}-\frac{8619}{8}a^{12}+\frac{31603}{16}a^{10}-\frac{9295}{4}a^{8}+\frac{13013}{8}a^{6}-\frac{1183}{2}a^{4}+\frac{169}{2}a^{2}-2$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}-\frac{25193}{256}a^{16}+\frac{12593}{32}a^{14}-\frac{4303}{4}a^{12}+\frac{31427}{16}a^{10}-\frac{4565}{2}a^{8}+\frac{12341}{8}a^{6}-\frac{1015}{2}a^{4}+\frac{105}{2}a^{2}$, $\frac{1}{131072}a^{34}-\frac{35}{65536}a^{32}+\frac{559}{32768}a^{30}-\frac{2697}{8192}a^{28}+\frac{8773}{2048}a^{26}-\frac{5083}{128}a^{24}+\frac{69181}{256}a^{22}-\frac{87725}{64}a^{20}+\frac{2666707}{512}a^{18}-\frac{3783145}{256}a^{16}+\frac{3962037}{128}a^{14}-\frac{1500063}{32}a^{12}+\frac{198575}{4}a^{10}-\frac{557883}{16}a^{8}+\frac{29727}{2}a^{6}-\frac{12981}{4}a^{4}+\frac{469}{2}a^{2}-2$, $\frac{1}{131072}a^{34}-\frac{35}{65536}a^{32}+\frac{559}{32768}a^{30}-\frac{5395}{16384}a^{28}+\frac{35119}{8192}a^{26}-\frac{40745}{1024}a^{24}+\frac{277863}{1024}a^{22}-\frac{11047}{8}a^{20}+\frac{2699215}{512}a^{18}-\frac{3853865}{256}a^{16}+\frac{4069629}{128}a^{14}-\frac{3113265}{64}a^{12}+\frac{208567}{4}a^{10}-\frac{296955}{8}a^{8}+\frac{128331}{8}a^{6}-\frac{14155}{4}a^{4}+254a^{2}-2$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{119}{16}a^{15}+\frac{735}{16}a^{13}-\frac{5733}{32}a^{11}+\frac{7007}{16}a^{9}-\frac{1287}{2}a^{7}+\frac{2079}{4}a^{5}-\frac{385}{2}a^{3}+21a-1$, $\frac{1}{16384}a^{28}-\frac{7}{2048}a^{26}+\frac{175}{2048}a^{24}-\frac{161}{128}a^{22}+\frac{1}{1024}a^{21}+\frac{12397}{1024}a^{20}-\frac{21}{512}a^{19}-\frac{10241}{128}a^{18}+\frac{189}{256}a^{17}+\frac{47481}{128}a^{16}-\frac{119}{16}a^{15}-\frac{4845}{4}a^{14}+\frac{735}{16}a^{13}+\frac{88179}{32}a^{12}-\frac{5733}{32}a^{11}-\frac{17017}{4}a^{10}+\frac{7007}{16}a^{9}+\frac{17017}{4}a^{8}-\frac{1287}{2}a^{7}-2548a^{6}+\frac{2079}{4}a^{5}+\frac{3185}{4}a^{4}-\frac{385}{2}a^{3}-98a^{2}+21a+2$, $\frac{1}{16384}a^{28}-\frac{7}{2048}a^{26}+\frac{175}{2048}a^{24}-\frac{161}{128}a^{22}+\frac{1}{1024}a^{21}+\frac{12397}{1024}a^{20}-\frac{21}{512}a^{19}-\frac{10241}{128}a^{18}+\frac{189}{256}a^{17}+\frac{47481}{128}a^{16}-\frac{119}{16}a^{15}-\frac{155039}{128}a^{14}+\frac{735}{16}a^{13}+\frac{22043}{8}a^{12}-\frac{5733}{32}a^{11}-\frac{136059}{32}a^{10}+\frac{7007}{16}a^{9}+\frac{33929}{8}a^{8}-\frac{1287}{2}a^{7}-\frac{10045}{4}a^{6}+\frac{2079}{4}a^{5}+\frac{2989}{4}a^{4}-\frac{385}{2}a^{3}-\frac{147}{2}a^{2}+21a$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{119}{16}a^{15}+\frac{735}{16}a^{13}-\frac{5733}{32}a^{11}+\frac{7007}{16}a^{9}-\frac{1287}{2}a^{7}+\frac{2079}{4}a^{5}-\frac{1}{4}a^{4}-\frac{385}{2}a^{3}+2a^{2}+21a-2$, $\frac{1}{131072}a^{34}-\frac{35}{65536}a^{32}+\frac{559}{32768}a^{30}-\frac{5395}{16384}a^{28}+\frac{35119}{8192}a^{26}-\frac{40745}{1024}a^{24}+\frac{277863}{1024}a^{22}+\frac{1}{1024}a^{21}-\frac{11047}{8}a^{20}-\frac{21}{512}a^{19}+\frac{2699215}{512}a^{18}+\frac{189}{256}a^{17}-\frac{3853865}{256}a^{16}-\frac{119}{16}a^{15}+\frac{4069629}{128}a^{14}+\frac{735}{16}a^{13}-\frac{3113265}{64}a^{12}-\frac{5733}{32}a^{11}+\frac{208567}{4}a^{10}+\frac{7007}{16}a^{9}-\frac{296955}{8}a^{8}-\frac{1287}{2}a^{7}+\frac{128331}{8}a^{6}+\frac{2079}{4}a^{5}-\frac{14155}{4}a^{4}-\frac{385}{2}a^{3}+254a^{2}+21a-2$, $\frac{1}{131072}a^{34}-\frac{35}{65536}a^{32}+\frac{559}{32768}a^{30}-\frac{5395}{16384}a^{28}+\frac{35119}{8192}a^{26}-\frac{40745}{1024}a^{24}+\frac{277863}{1024}a^{22}-\frac{1}{1024}a^{21}-\frac{11047}{8}a^{20}+\frac{21}{512}a^{19}+\frac{2699215}{512}a^{18}-\frac{189}{256}a^{17}-\frac{3853865}{256}a^{16}+\frac{119}{16}a^{15}+\frac{4069629}{128}a^{14}-\frac{735}{16}a^{13}-\frac{3113265}{64}a^{12}+\frac{5733}{32}a^{11}+\frac{208567}{4}a^{10}-\frac{7007}{16}a^{9}-\frac{296955}{8}a^{8}+\frac{1287}{2}a^{7}+\frac{128331}{8}a^{6}-\frac{2079}{4}a^{5}-\frac{7077}{2}a^{4}+\frac{385}{2}a^{3}+252a^{2}-21a$, $\frac{1}{131072}a^{34}-\frac{35}{65536}a^{32}+\frac{559}{32768}a^{30}-\frac{2697}{8192}a^{28}+\frac{8773}{2048}a^{26}-\frac{5083}{128}a^{24}+\frac{69181}{256}a^{22}-\frac{1}{1024}a^{21}-\frac{87725}{64}a^{20}+\frac{21}{512}a^{19}+\frac{2666707}{512}a^{18}-\frac{189}{256}a^{17}-\frac{3783145}{256}a^{16}+\frac{119}{16}a^{15}+\frac{3962037}{128}a^{14}-\frac{735}{16}a^{13}-\frac{1500063}{32}a^{12}+\frac{5733}{32}a^{11}+\frac{198575}{4}a^{10}-\frac{7007}{16}a^{9}-\frac{557883}{16}a^{8}+\frac{1287}{2}a^{7}+\frac{29727}{2}a^{6}-\frac{2079}{4}a^{5}-\frac{12981}{4}a^{4}+\frac{385}{2}a^{3}+\frac{469}{2}a^{2}-21a-2$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{119}{16}a^{15}+\frac{735}{16}a^{13}-\frac{5733}{32}a^{11}+\frac{7007}{16}a^{9}-\frac{1287}{2}a^{7}+\frac{2079}{4}a^{5}-\frac{385}{2}a^{3}-\frac{1}{2}a^{2}+21a+2$, $\frac{1}{2048}a^{22}-\frac{1}{1024}a^{21}-\frac{21}{1024}a^{20}+\frac{21}{512}a^{19}+\frac{189}{512}a^{18}-\frac{189}{256}a^{17}-\frac{119}{32}a^{16}+\frac{119}{16}a^{15}+\frac{735}{32}a^{14}-\frac{735}{16}a^{13}-\frac{5733}{64}a^{12}+\frac{5733}{32}a^{11}+\frac{7007}{32}a^{10}-\frac{7007}{16}a^{9}-\frac{1287}{4}a^{8}+\frac{1287}{2}a^{7}+\frac{2079}{8}a^{6}-\frac{2079}{4}a^{5}-\frac{385}{4}a^{4}+\frac{385}{2}a^{3}+\frac{21}{2}a^{2}-21a$, $\frac{1}{1024}a^{21}+\frac{1}{1024}a^{20}-\frac{21}{512}a^{19}-\frac{5}{128}a^{18}+\frac{189}{256}a^{17}+\frac{85}{128}a^{16}-\frac{119}{16}a^{15}-\frac{25}{4}a^{14}+\frac{735}{16}a^{13}+\frac{2275}{64}a^{12}-\frac{5733}{32}a^{11}-\frac{1001}{8}a^{10}+\frac{7007}{16}a^{9}+\frac{2145}{8}a^{8}-\frac{1287}{2}a^{7}-330a^{6}+\frac{2079}{4}a^{5}+\frac{825}{4}a^{4}-\frac{385}{2}a^{3}-50a^{2}+21a+2$, $\frac{1}{65536}a^{32}-\frac{1}{1024}a^{30}+\frac{29}{1024}a^{28}-\frac{63}{128}a^{26}+\frac{2925}{512}a^{24}-\frac{1495}{32}a^{22}-\frac{1}{1024}a^{21}+\frac{8855}{32}a^{20}+\frac{21}{512}a^{19}-\frac{4807}{4}a^{18}-\frac{189}{256}a^{17}+\frac{245157}{64}a^{16}+\frac{119}{16}a^{15}-\frac{17765}{2}a^{14}-\frac{735}{16}a^{13}+\frac{29393}{2}a^{12}+\frac{5733}{32}a^{11}-\frac{537471}{32}a^{10}-\frac{7007}{16}a^{9}+\frac{100771}{8}a^{8}+\frac{1287}{2}a^{7}-\frac{45661}{8}a^{6}-\frac{2079}{4}a^{5}+\frac{2695}{2}a^{4}+\frac{385}{2}a^{3}-\frac{231}{2}a^{2}-21a$, $\frac{1}{131072}a^{34}-\frac{17}{32768}a^{32}+\frac{527}{32768}a^{30}-\frac{2465}{8192}a^{28}+\frac{31059}{8192}a^{26}-\frac{69615}{2048}a^{24}+\frac{228735}{1024}a^{22}+\frac{1}{1024}a^{21}-\frac{279565}{256}a^{20}-\frac{21}{512}a^{19}+\frac{2042975}{512}a^{18}+\frac{189}{256}a^{17}-\frac{1389223}{128}a^{16}-\frac{119}{16}a^{15}+\frac{1389223}{64}a^{14}+\frac{735}{16}a^{13}-\frac{499681}{16}a^{12}-\frac{5733}{32}a^{11}+\frac{499681}{16}a^{10}+\frac{7007}{16}a^{9}-\frac{82365}{4}a^{8}-\frac{1287}{2}a^{7}+\frac{16473}{2}a^{6}+\frac{2079}{4}a^{5}-1734a^{4}-\frac{385}{2}a^{3}+\frac{289}{2}a^{2}+21a-2$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{1}{512}a^{18}+\frac{189}{256}a^{17}-\frac{9}{128}a^{16}-\frac{119}{16}a^{15}+\frac{135}{128}a^{14}+\frac{735}{16}a^{13}-\frac{273}{32}a^{12}-\frac{5733}{32}a^{11}+\frac{1287}{32}a^{10}+\frac{7007}{16}a^{9}-\frac{891}{8}a^{8}-\frac{1287}{2}a^{7}+\frac{693}{4}a^{6}+\frac{2079}{4}a^{5}-135a^{4}-\frac{385}{2}a^{3}+\frac{81}{2}a^{2}+21a-2$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{119}{16}a^{15}+\frac{735}{16}a^{13}-\frac{5733}{32}a^{11}+\frac{1}{32}a^{10}+\frac{7007}{16}a^{9}-\frac{5}{8}a^{8}-\frac{1287}{2}a^{7}+\frac{35}{8}a^{6}+\frac{2079}{4}a^{5}-\frac{25}{2}a^{4}-\frac{385}{2}a^{3}+\frac{25}{2}a^{2}+21a-2$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{119}{16}a^{15}+\frac{735}{16}a^{13}-\frac{5733}{32}a^{11}+\frac{7007}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1287}{2}a^{7}-a^{6}+\frac{2079}{4}a^{5}+5a^{4}-\frac{385}{2}a^{3}-8a^{2}+21a+2$, $\frac{1}{32768}a^{30}-\frac{15}{8192}a^{28}+\frac{405}{8192}a^{26}-\frac{1625}{2048}a^{24}+\frac{8625}{1024}a^{22}+\frac{1}{1024}a^{21}-\frac{15939}{256}a^{20}-\frac{21}{512}a^{19}+\frac{42061}{128}a^{18}+\frac{189}{256}a^{17}-\frac{39969}{32}a^{16}-\frac{119}{16}a^{15}+\frac{435915}{128}a^{14}+\frac{735}{16}a^{13}-\frac{419353}{64}a^{12}-\frac{5733}{32}a^{11}+\frac{275835}{32}a^{10}+\frac{3503}{8}a^{9}-7344a^{8}-\frac{5139}{8}a^{7}+\frac{14721}{4}a^{6}+513a^{5}-\frac{3555}{4}a^{4}-\frac{355}{2}a^{3}+54a^{2}+12a$, $\frac{1}{32768}a^{30}-\frac{15}{8192}a^{28}+\frac{405}{8192}a^{26}-\frac{1625}{2048}a^{24}+\frac{8625}{1024}a^{22}-\frac{1}{1024}a^{21}-\frac{15939}{256}a^{20}+\frac{21}{512}a^{19}+\frac{42061}{128}a^{18}-\frac{189}{256}a^{17}-\frac{39969}{32}a^{16}+\frac{119}{16}a^{15}+\frac{435915}{128}a^{14}-\frac{735}{16}a^{13}-\frac{419353}{64}a^{12}+\frac{5733}{32}a^{11}+\frac{275835}{32}a^{10}-\frac{7007}{16}a^{9}-7344a^{8}+\frac{1287}{2}a^{7}+\frac{29443}{8}a^{6}-\frac{2079}{4}a^{5}-\frac{3561}{4}a^{4}+\frac{385}{2}a^{3}+\frac{117}{2}a^{2}-21a-1$, $\frac{1}{2048}a^{22}-\frac{1}{1024}a^{21}-\frac{11}{512}a^{20}+\frac{21}{512}a^{19}+\frac{209}{512}a^{18}-\frac{189}{256}a^{17}-\frac{561}{128}a^{16}+\frac{119}{16}a^{15}+\frac{935}{32}a^{14}-\frac{735}{16}a^{13}-\frac{1001}{8}a^{12}+\frac{5733}{32}a^{11}+\frac{11011}{32}a^{10}-\frac{7007}{16}a^{9}-\frac{4719}{8}a^{8}+\frac{1287}{2}a^{7}+\frac{4719}{8}a^{6}-\frac{2079}{4}a^{5}-\frac{605}{2}a^{4}+\frac{385}{2}a^{3}+\frac{121}{2}a^{2}-21a-2$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 19433880349688750000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{36}\cdot(2\pi)^{0}\cdot 19433880349688750000000 \cdot 1}{2\cdot\sqrt{23610692285332399309092778573219694177406932269512586359440570056704}}\cr\approx \mathstrut & 0.137421603633658 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 72*x^34 + 2376*x^32 - 47624*x^30 + 647760*x^28 - 6327072*x^26 + 45809920*x^24 - 250210560*x^22 + 1039332096*x^20 - 3285680128*x^18 + 7858649088*x^16 - 14038161408*x^14 + 18332004352*x^12 - 16929153024*x^10 + 10498277376*x^8 - 4012965888*x^6 + 804519936*x^4 - 56623104*x^2 + 262144);
 
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

$C_6^2$ (as 36T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\sqrt{2}, \sqrt{21})\), 6.6.3359232.1, 6.6.1229312.1, 6.6.8065516032.1, 6.6.8065516032.2, 6.6.3456649728.1, 6.6.6751269.1, 6.6.232339968.1, \(\Q(\zeta_{21})^+\), 6.6.169375836672.1, 6.6.330812181.2, 6.6.169375836672.2, 6.6.330812181.1, 9.9.62523502209.1, 12.12.11948427342082473984.2, 12.12.53981860730241024.1, 12.12.28688174048340020035584.2, 12.12.28688174048340020035584.4, 18.18.524682375772545974113841184768.1, 18.18.4859083482029548266268283212136448.1, \(\Q(\zeta_{63})^+\)

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-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 R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.3.0.1}{3} }^{12}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{12}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display 7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$
7.18.15.5$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$$6$$3$$15$$C_6 \times C_3$$[\ ]_{6}^{3}$