Properties

Label 36.0.204...304.1
Degree $36$
Signature $(0, 18)$
Discriminant $2.044\times 10^{79}$
Root discriminant \(159.61\)
Ramified primes $2,7,13,47,67$
Class number not computed
Class group not computed
Galois group $A_4^2:C_2^3$ (as 36T1803)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392)
 
Copy content gp:K = bnfinit(y^36 + 13*y^34 - 26*y^33 + 94*y^32 - 262*y^31 + 731*y^30 - 1205*y^29 + 4274*y^28 - 1133*y^27 + 6981*y^26 - 11062*y^25 - 109813*y^24 - 65593*y^23 - 338726*y^22 - 14803*y^21 + 1910654*y^20 + 3560178*y^19 + 19134908*y^18 + 11250407*y^17 + 41426055*y^16 + 87117071*y^15 - 47139783*y^14 + 24379819*y^13 + 323548427*y^12 - 115066148*y^11 + 662418*y^10 + 484355456*y^9 - 417892300*y^8 - 73724176*y^7 + 692053512*y^6 - 56244592*y^5 - 152888384*y^4 + 87904448*y^3 + 14238336*y^2 - 7037952*y + 3195392, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392)
 

\( x^{36} + 13 x^{34} - 26 x^{33} + 94 x^{32} - 262 x^{31} + 731 x^{30} - 1205 x^{29} + 4274 x^{28} + \cdots + 3195392 \) 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:  $36$
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:  $(0, 18)$
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:   \(20437599181597888521607217689901451581497256981403444028932632651429364011106304\) \(\medspace = 2^{18}\cdot 7^{18}\cdot 13^{24}\cdot 47^{6}\cdot 67^{12}\) 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:  \(159.61\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}7^{3/4}13^{2/3}47^{1/2}67^{1/2}\approx 3776.450141979516$
Ramified primes:   \(2\), \(7\), \(13\), \(47\), \(67\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{18}-\frac{1}{2}a^{14}-\frac{1}{6}a^{12}-\frac{1}{2}a^{11}+\frac{1}{3}a^{10}-\frac{1}{2}a^{7}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{19}-\frac{1}{2}a^{15}-\frac{1}{6}a^{13}-\frac{1}{2}a^{12}+\frac{1}{3}a^{11}-\frac{1}{2}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{20}+\frac{1}{3}a^{14}-\frac{1}{2}a^{13}+\frac{1}{3}a^{12}-\frac{1}{2}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}$, $\frac{1}{6}a^{21}+\frac{1}{3}a^{15}-\frac{1}{2}a^{14}+\frac{1}{3}a^{13}-\frac{1}{2}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}$, $\frac{1}{12}a^{22}-\frac{1}{12}a^{20}-\frac{1}{12}a^{18}-\frac{1}{4}a^{17}+\frac{1}{6}a^{16}-\frac{1}{2}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{6}a^{12}-\frac{1}{2}a^{11}+\frac{1}{6}a^{10}+\frac{1}{4}a^{9}+\frac{1}{12}a^{8}+\frac{1}{4}a^{7}-\frac{1}{6}a^{6}-\frac{1}{4}a^{5}-\frac{1}{3}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{12}a^{23}-\frac{1}{12}a^{21}-\frac{1}{12}a^{19}-\frac{1}{12}a^{18}+\frac{1}{6}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{1}{6}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{12}a^{10}-\frac{5}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{6}a^{7}+\frac{5}{12}a^{6}+\frac{1}{6}a^{5}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{24}-\frac{1}{12}a^{19}-\frac{1}{12}a^{18}-\frac{1}{4}a^{17}-\frac{1}{12}a^{16}+\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{5}{12}a^{13}+\frac{1}{3}a^{12}+\frac{1}{12}a^{11}-\frac{1}{12}a^{10}-\frac{5}{12}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{5}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{12}a^{25}-\frac{1}{12}a^{20}-\frac{1}{12}a^{19}-\frac{1}{12}a^{18}-\frac{1}{12}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{5}{12}a^{14}+\frac{1}{3}a^{13}-\frac{1}{12}a^{12}+\frac{5}{12}a^{11}-\frac{1}{6}a^{10}+\frac{1}{12}a^{9}-\frac{1}{3}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{12}a^{26}-\frac{1}{12}a^{21}-\frac{1}{12}a^{20}-\frac{1}{12}a^{19}-\frac{1}{12}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{5}{12}a^{15}+\frac{1}{3}a^{14}-\frac{1}{12}a^{13}+\frac{5}{12}a^{12}-\frac{1}{6}a^{11}+\frac{1}{12}a^{10}-\frac{1}{3}a^{9}+\frac{1}{6}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a$, $\frac{1}{12}a^{27}-\frac{1}{12}a^{21}-\frac{1}{12}a^{19}-\frac{1}{4}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{12}a^{11}-\frac{1}{2}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{24}a^{28}-\frac{1}{24}a^{26}-\frac{1}{24}a^{22}-\frac{1}{24}a^{21}-\frac{1}{12}a^{20}-\frac{1}{24}a^{19}+\frac{1}{24}a^{18}+\frac{1}{24}a^{16}+\frac{7}{24}a^{15}-\frac{1}{2}a^{14}+\frac{5}{24}a^{13}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}+\frac{5}{12}a^{10}-\frac{1}{24}a^{9}-\frac{1}{24}a^{8}+\frac{1}{8}a^{7}-\frac{1}{24}a^{6}-\frac{1}{8}a^{5}-\frac{5}{24}a^{4}-\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{3}a$, $\frac{1}{24}a^{29}-\frac{1}{24}a^{27}-\frac{1}{24}a^{23}-\frac{1}{24}a^{22}-\frac{1}{12}a^{21}-\frac{1}{24}a^{20}+\frac{1}{24}a^{19}+\frac{1}{24}a^{17}-\frac{5}{24}a^{16}-\frac{1}{2}a^{15}-\frac{7}{24}a^{14}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{5}{12}a^{11}+\frac{11}{24}a^{10}+\frac{11}{24}a^{9}-\frac{3}{8}a^{8}+\frac{11}{24}a^{7}+\frac{3}{8}a^{6}-\frac{5}{24}a^{5}-\frac{1}{4}a^{4}-\frac{5}{12}a^{3}+\frac{1}{6}a^{2}$, $\frac{1}{144}a^{30}-\frac{1}{72}a^{29}-\frac{1}{144}a^{28}+\frac{1}{36}a^{27}-\frac{1}{24}a^{25}+\frac{1}{48}a^{24}-\frac{1}{48}a^{23}-\frac{1}{24}a^{22}-\frac{7}{144}a^{21}-\frac{1}{144}a^{20}+\frac{5}{72}a^{19}-\frac{7}{144}a^{18}-\frac{1}{48}a^{17}+\frac{1}{8}a^{16}-\frac{41}{144}a^{15}-\frac{5}{36}a^{14}-\frac{1}{2}a^{13}+\frac{17}{36}a^{12}-\frac{37}{144}a^{11}+\frac{49}{144}a^{10}-\frac{61}{144}a^{9}+\frac{17}{144}a^{8}+\frac{35}{144}a^{7}-\frac{11}{48}a^{6}-\frac{2}{9}a^{5}+\frac{1}{18}a^{4}-\frac{11}{36}a^{3}+\frac{1}{36}a^{2}+\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{288}a^{31}-\frac{5}{288}a^{29}+\frac{1}{144}a^{28}+\frac{1}{36}a^{27}-\frac{1}{48}a^{26}+\frac{1}{96}a^{25}-\frac{1}{32}a^{24}-\frac{1}{24}a^{23}-\frac{7}{288}a^{22}-\frac{5}{96}a^{21}+\frac{1}{36}a^{20}-\frac{11}{288}a^{19}+\frac{19}{288}a^{18}+\frac{67}{288}a^{16}+\frac{19}{48}a^{15}+\frac{23}{72}a^{14}-\frac{13}{72}a^{13}-\frac{1}{32}a^{12}-\frac{97}{288}a^{11}+\frac{1}{288}a^{10}-\frac{43}{96}a^{9}+\frac{5}{32}a^{8}-\frac{23}{288}a^{7}-\frac{61}{144}a^{6}+\frac{7}{72}a^{5}+\frac{5}{18}a^{4}-\frac{5}{24}a^{3}+\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{4}{9}$, $\frac{1}{576}a^{32}-\frac{1}{576}a^{30}+\frac{1}{96}a^{29}+\frac{1}{144}a^{28}+\frac{11}{288}a^{27}-\frac{7}{192}a^{26}-\frac{1}{64}a^{25}-\frac{1}{24}a^{24}+\frac{17}{576}a^{23}-\frac{1}{192}a^{22}+\frac{1}{144}a^{21}-\frac{3}{64}a^{20}-\frac{25}{576}a^{19}-\frac{1}{144}a^{18}+\frac{139}{576}a^{17}+\frac{5}{96}a^{16}-\frac{5}{12}a^{15}+\frac{7}{24}a^{14}-\frac{43}{192}a^{13}+\frac{55}{576}a^{12}-\frac{49}{192}a^{11}+\frac{55}{576}a^{10}-\frac{115}{576}a^{9}-\frac{7}{64}a^{8}+\frac{15}{32}a^{7}-\frac{53}{144}a^{6}-\frac{7}{16}a^{5}+\frac{17}{144}a^{4}-\frac{31}{72}a^{3}-\frac{5}{36}a^{2}+\frac{4}{9}$, $\frac{1}{576}a^{33}-\frac{1}{576}a^{31}-\frac{1}{288}a^{30}-\frac{1}{144}a^{29}+\frac{1}{96}a^{28}+\frac{19}{576}a^{27}+\frac{5}{192}a^{26}-\frac{1}{24}a^{25}-\frac{7}{576}a^{24}-\frac{1}{192}a^{23}+\frac{1}{144}a^{22}+\frac{5}{576}a^{21}+\frac{7}{576}a^{20}-\frac{1}{16}a^{19}-\frac{7}{192}a^{18}-\frac{1}{32}a^{17}+\frac{1}{6}a^{16}+\frac{5}{72}a^{15}-\frac{89}{576}a^{14}+\frac{175}{576}a^{13}+\frac{221}{576}a^{12}+\frac{53}{192}a^{11}-\frac{65}{192}a^{10}-\frac{55}{576}a^{9}-\frac{5}{288}a^{8}-\frac{1}{48}a^{7}+\frac{1}{48}a^{6}-\frac{3}{16}a^{5}+\frac{1}{12}a^{4}-\frac{4}{9}a^{3}-\frac{11}{36}a^{2}+\frac{1}{18}a-\frac{2}{9}$, $\frac{1}{1152}a^{34}-\frac{1}{1152}a^{32}-\frac{1}{576}a^{31}-\frac{1}{288}a^{30}-\frac{1}{64}a^{29}+\frac{19}{1152}a^{28}-\frac{1}{128}a^{27}+\frac{1}{48}a^{26}-\frac{7}{1152}a^{25}+\frac{5}{128}a^{24}-\frac{5}{288}a^{23}+\frac{29}{1152}a^{22}+\frac{7}{1152}a^{21}-\frac{5}{96}a^{20}-\frac{5}{128}a^{19}+\frac{5}{192}a^{18}+\frac{11}{48}a^{17}-\frac{11}{72}a^{16}-\frac{137}{1152}a^{15}+\frac{247}{1152}a^{14}+\frac{461}{1152}a^{13}-\frac{25}{128}a^{12}+\frac{79}{384}a^{11}-\frac{175}{1152}a^{10}-\frac{137}{576}a^{9}+\frac{5}{96}a^{8}+\frac{5}{32}a^{7}-\frac{19}{96}a^{6}+\frac{19}{48}a^{5}-\frac{2}{9}a^{4}-\frac{5}{18}a^{3}-\frac{1}{18}a^{2}-\frac{4}{9}a$, $\frac{1}{86\cdots 08}a^{35}-\frac{45\cdots 97}{54\cdots 76}a^{34}+\frac{60\cdots 01}{86\cdots 08}a^{33}-\frac{10\cdots 85}{36\cdots 92}a^{32}-\frac{20\cdots 97}{43\cdots 04}a^{31}-\frac{34\cdots 63}{14\cdots 68}a^{30}+\frac{68\cdots 63}{86\cdots 08}a^{29}-\frac{58\cdots 81}{28\cdots 36}a^{28}-\frac{44\cdots 53}{21\cdots 52}a^{27}-\frac{78\cdots 61}{86\cdots 08}a^{26}+\frac{24\cdots 39}{86\cdots 08}a^{25}+\frac{81\cdots 63}{10\cdots 76}a^{24}+\frac{48\cdots 45}{28\cdots 36}a^{23}+\frac{10\cdots 69}{86\cdots 08}a^{22}-\frac{11\cdots 05}{24\cdots 28}a^{21}-\frac{53\cdots 17}{28\cdots 36}a^{20}-\frac{14\cdots 47}{10\cdots 76}a^{19}-\frac{10\cdots 95}{43\cdots 04}a^{18}-\frac{31\cdots 37}{13\cdots 72}a^{17}+\frac{13\cdots 55}{96\cdots 12}a^{16}-\frac{97\cdots 47}{96\cdots 12}a^{15}-\frac{43\cdots 43}{86\cdots 08}a^{14}+\frac{10\cdots 27}{86\cdots 08}a^{13}-\frac{28\cdots 83}{86\cdots 08}a^{12}+\frac{30\cdots 53}{86\cdots 08}a^{11}+\frac{14\cdots 43}{43\cdots 04}a^{10}+\frac{42\cdots 09}{14\cdots 68}a^{9}-\frac{58\cdots 29}{24\cdots 28}a^{8}-\frac{69\cdots 79}{72\cdots 84}a^{7}-\frac{10\cdots 43}{23\cdots 08}a^{6}-\frac{42\cdots 37}{10\cdots 76}a^{5}+\frac{21\cdots 47}{13\cdots 72}a^{4}+\frac{26\cdots 07}{17\cdots 68}a^{3}+\frac{26\cdots 29}{11\cdots 31}a^{2}-\frac{16\cdots 63}{67\cdots 86}a-\frac{15\cdots 45}{42\cdots 67}$ 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:  $3$

Class group and class number

Ideal class group:  not computed
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:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   data not computed

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:  $17$
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:   \( -1 \)  (order $2$) 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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot R \cdot h}{2\cdot\sqrt{20437599181597888521607217689901451581497256981403444028932632651429364011106304}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392) 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^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392, 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^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392); 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 = polynomial_ring(QQ); K, a = number_field(x^36 + 13*x^34 - 26*x^33 + 94*x^32 - 262*x^31 + 731*x^30 - 1205*x^29 + 4274*x^28 - 1133*x^27 + 6981*x^26 - 11062*x^25 - 109813*x^24 - 65593*x^23 - 338726*x^22 - 14803*x^21 + 1910654*x^20 + 3560178*x^19 + 19134908*x^18 + 11250407*x^17 + 41426055*x^16 + 87117071*x^15 - 47139783*x^14 + 24379819*x^13 + 323548427*x^12 - 115066148*x^11 + 662418*x^10 + 484355456*x^9 - 417892300*x^8 - 73724176*x^7 + 692053512*x^6 - 56244592*x^5 - 152888384*x^4 + 87904448*x^3 + 14238336*x^2 - 7037952*x + 3195392); 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

$A_4^2:C_2^3$ (as 36T1803):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 1152
The 80 conjugacy class representatives for $A_4^2:C_2^3$
Character table for $A_4^2:C_2^3$

Intermediate fields

3.3.169.1, 3.3.469.1, 6.0.12317816.1, 6.0.1342367.1, 9.9.497941865456581.1, 18.0.43543302538963085509483263938264576.1, 18.0.25742508082992235484529205265251703.1, 18.18.4520796299502764426851080911862443074048.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 36 siblings: deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, some data not computed
Minimal sibling: not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{12}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{8}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ R ${\href{/padicField/17.12.0.1}{12} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.6.0.1}{6} }^{2}$ R ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.3.2.9a1.5$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$3$$9$$C_6$$$[3]^{3}$$
2.3.2.9a1.5$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 7$$2$$3$$9$$C_6$$$[3]^{3}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(7\) Copy content Toggle raw display 7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.4.9a1.1$x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1584 x^{8} + 1728 x^{7} + 3552 x^{6} + 1152 x^{5} + 3456 x^{4} + 256 x^{3} + 1536 x^{2} + 7 x + 256$$4$$3$$9$$D_4 \times C_3$$$[\ ]_{4}^{6}$$
7.3.4.9a1.1$x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1584 x^{8} + 1728 x^{7} + 3552 x^{6} + 1152 x^{5} + 3456 x^{4} + 256 x^{3} + 1536 x^{2} + 7 x + 256$$4$$3$$9$$D_4 \times C_3$$$[\ ]_{4}^{6}$$
\(13\) Copy content Toggle raw display 13.6.3.12a1.3$x^{18} + 30 x^{15} + 33 x^{14} + 33 x^{13} + 306 x^{12} + 660 x^{11} + 1023 x^{10} + 1846 x^{9} + 3795 x^{8} + 7062 x^{7} + 9203 x^{6} + 8943 x^{5} + 6039 x^{4} + 2903 x^{3} + 858 x^{2} + 132 x + 21$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$
13.6.3.12a1.3$x^{18} + 30 x^{15} + 33 x^{14} + 33 x^{13} + 306 x^{12} + 660 x^{11} + 1023 x^{10} + 1846 x^{9} + 3795 x^{8} + 7062 x^{7} + 9203 x^{6} + 8943 x^{5} + 6039 x^{4} + 2903 x^{3} + 858 x^{2} + 132 x + 21$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.1.0a1.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
47.4.2.4a1.2$x^{8} + 16 x^{6} + 80 x^{5} + 74 x^{4} + 640 x^{3} + 1680 x^{2} + 400 x + 72$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(67\) Copy content Toggle raw display 67.6.1.0a1.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
67.6.1.0a1.1$x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
67.6.2.6a1.2$x^{12} + 126 x^{9} + 98 x^{8} + 110 x^{7} + 3973 x^{6} + 6174 x^{5} + 9331 x^{4} + 5642 x^{3} + 3221 x^{2} + 220 x + 71$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
67.6.2.6a1.2$x^{12} + 126 x^{9} + 98 x^{8} + 110 x^{7} + 3973 x^{6} + 6174 x^{5} + 9331 x^{4} + 5642 x^{3} + 3221 x^{2} + 220 x + 71$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$

Spectrum of ring of integers

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