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Properties

Label 28.4.471...816.1
Degree $28$
Signature $[4, 12]$
Discriminant $4.718\times 10^{57}$
Root discriminant \(114.76\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $G(2,2)$ (as 28T393)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600)
 
Copy content gp:K = bnfinit(y^28 + 18*y^26 + 225*y^24 - 288*y^23 + 1152*y^22 - 5760*y^21 + 18108*y^20 - 89952*y^19 + 217224*y^18 - 499680*y^17 + 2230524*y^16 - 814464*y^15 + 13974624*y^14 + 12253824*y^13 + 48314916*y^12 + 71792064*y^11 + 68082600*y^10 + 88956864*y^9 + 11560068*y^8 - 83134080*y^7 - 93788352*y^6 - 73624320*y^5 - 34564032*y^4 - 3382272*y^3 + 1400832*y^2 + 22528*y + 57600, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600)
 

\( x^{28} + 18 x^{26} + 225 x^{24} - 288 x^{23} + 1152 x^{22} - 5760 x^{21} + 18108 x^{20} - 89952 x^{19} + \cdots + 57600 \) 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:  $28$
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:  $[4, 12]$
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:   \(4717667136961114442493020206737586382842235909528214306816\) \(\medspace = 2^{106}\cdot 3^{54}\) 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:  \(114.76\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{18}a^{10}+\frac{1}{3}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{9}a$, $\frac{1}{18}a^{11}-\frac{1}{6}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{36}a^{12}-\frac{1}{36}a^{10}-\frac{1}{3}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{6}a^{4}-\frac{2}{9}a^{3}+\frac{1}{6}a^{2}-\frac{1}{9}a$, $\frac{1}{36}a^{13}-\frac{1}{36}a^{11}+\frac{1}{6}a^{9}+\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{2}{9}a^{4}+\frac{1}{6}a^{3}-\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{72}a^{14}-\frac{1}{72}a^{13}-\frac{1}{72}a^{12}-\frac{1}{72}a^{11}-\frac{1}{36}a^{10}+\frac{1}{3}a^{9}+\frac{1}{12}a^{6}-\frac{7}{36}a^{5}-\frac{11}{36}a^{4}+\frac{1}{36}a^{3}-\frac{7}{18}a^{2}+\frac{2}{9}a$, $\frac{1}{72}a^{15}-\frac{1}{72}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{5}{12}a^{7}-\frac{4}{9}a^{6}+\frac{1}{3}a^{4}+\frac{1}{4}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{144}a^{16}-\frac{1}{72}a^{13}-\frac{1}{144}a^{12}-\frac{1}{72}a^{11}+\frac{1}{3}a^{9}+\frac{3}{8}a^{8}+\frac{4}{9}a^{7}-\frac{1}{2}a^{6}-\frac{1}{12}a^{5}-\frac{31}{72}a^{4}-\frac{7}{36}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{144}a^{17}+\frac{1}{144}a^{13}+\frac{1}{72}a^{11}-\frac{7}{24}a^{9}+\frac{1}{9}a^{8}-\frac{1}{6}a^{7}-\frac{1}{3}a^{6}-\frac{1}{8}a^{5}+\frac{1}{9}a^{4}+\frac{1}{12}a^{3}+\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{288}a^{18}+\frac{1}{288}a^{14}-\frac{1}{72}a^{13}+\frac{1}{144}a^{12}-\frac{1}{72}a^{11}+\frac{1}{48}a^{10}-\frac{4}{9}a^{9}+\frac{1}{12}a^{8}+\frac{7}{16}a^{6}-\frac{7}{36}a^{5}-\frac{1}{72}a^{4}+\frac{13}{36}a^{3}-\frac{1}{18}a^{2}-\frac{1}{3}a$, $\frac{1}{2592}a^{19}+\frac{1}{864}a^{18}-\frac{1}{432}a^{17}+\frac{1}{432}a^{16}-\frac{1}{864}a^{15}+\frac{5}{864}a^{14}-\frac{1}{216}a^{13}+\frac{1}{216}a^{12}-\frac{7}{432}a^{11}+\frac{25}{1296}a^{10}-\frac{65}{216}a^{9}-\frac{71}{216}a^{8}-\frac{167}{432}a^{7}-\frac{125}{432}a^{6}-\frac{35}{108}a^{5}-\frac{41}{108}a^{4}-\frac{23}{54}a^{3}-\frac{23}{54}a^{2}+\frac{35}{81}a-\frac{1}{9}$, $\frac{1}{2592}a^{20}+\frac{1}{864}a^{18}+\frac{1}{432}a^{17}-\frac{1}{864}a^{16}-\frac{1}{216}a^{15}-\frac{1}{864}a^{14}+\frac{5}{432}a^{13}-\frac{1}{108}a^{12}+\frac{1}{81}a^{11}+\frac{7}{432}a^{10}+\frac{103}{216}a^{9}+\frac{13}{432}a^{8}+\frac{43}{108}a^{7}-\frac{95}{432}a^{6}+\frac{83}{216}a^{5}-\frac{23}{216}a^{4}-\frac{4}{27}a^{3}+\frac{79}{162}a^{2}-\frac{11}{27}a+\frac{1}{3}$, $\frac{1}{2592}a^{21}-\frac{1}{864}a^{18}-\frac{1}{864}a^{17}+\frac{1}{432}a^{16}+\frac{1}{432}a^{15}-\frac{5}{864}a^{14}-\frac{1}{432}a^{13}+\frac{1}{81}a^{12}-\frac{1}{216}a^{11}+\frac{1}{432}a^{10}+\frac{169}{432}a^{9}+\frac{77}{216}a^{8}+\frac{71}{216}a^{7}+\frac{109}{432}a^{6}+\frac{35}{108}a^{5}-\frac{1}{27}a^{4}-\frac{31}{324}a^{3}+\frac{7}{27}a^{2}+\frac{1}{27}a+\frac{1}{3}$, $\frac{1}{2592}a^{22}-\frac{1}{864}a^{18}+\frac{1}{432}a^{17}+\frac{1}{432}a^{16}+\frac{1}{216}a^{15}-\frac{1}{432}a^{14}-\frac{11}{1296}a^{13}-\frac{1}{216}a^{12}-\frac{1}{216}a^{11}-\frac{7}{432}a^{10}+\frac{95}{216}a^{9}+\frac{71}{216}a^{8}+\frac{25}{108}a^{7}+\frac{35}{108}a^{6}-\frac{71}{216}a^{5}+\frac{77}{324}a^{4}-\frac{35}{108}a^{3}+\frac{7}{27}a^{2}-\frac{13}{27}a-\frac{1}{3}$, $\frac{1}{2592}a^{23}-\frac{1}{864}a^{18}+\frac{1}{432}a^{17}-\frac{1}{432}a^{16}-\frac{5}{864}a^{15}+\frac{5}{2592}a^{14}-\frac{5}{432}a^{13}+\frac{1}{108}a^{12}+\frac{1}{216}a^{11}+\frac{5}{432}a^{10}-\frac{103}{216}a^{9}+\frac{23}{216}a^{8}+\frac{47}{432}a^{7}-\frac{31}{432}a^{6}+\frac{19}{648}a^{5}+\frac{17}{54}a^{4}+\frac{43}{108}a^{3}+\frac{19}{54}a^{2}+\frac{5}{27}a-\frac{1}{3}$, $\frac{1}{20736}a^{24}+\frac{1}{10368}a^{22}+\frac{1}{20736}a^{20}-\frac{1}{864}a^{18}+\frac{1}{432}a^{17}+\frac{1}{1728}a^{16}-\frac{1}{324}a^{15}-\frac{1}{144}a^{14}-\frac{5}{1296}a^{13}+\frac{1}{192}a^{12}+\frac{1}{162}a^{11}+\frac{1}{216}a^{10}-\frac{19}{216}a^{9}+\frac{103}{1728}a^{8}-\frac{8}{27}a^{7}+\frac{371}{2592}a^{6}+\frac{5}{72}a^{5}+\frac{1865}{5184}a^{4}-\frac{1}{9}a^{3}-\frac{7}{648}a^{2}-\frac{17}{54}a-\frac{5}{24}$, $\frac{1}{20736}a^{25}+\frac{1}{10368}a^{23}+\frac{1}{20736}a^{21}-\frac{1}{864}a^{18}+\frac{1}{1728}a^{17}-\frac{1}{324}a^{16}+\frac{1}{288}a^{15}+\frac{17}{2592}a^{14}+\frac{7}{576}a^{13}+\frac{17}{1296}a^{12}+\frac{11}{432}a^{11}-\frac{7}{432}a^{10}-\frac{137}{1728}a^{9}-\frac{23}{108}a^{8}+\frac{1181}{2592}a^{7}+\frac{55}{144}a^{6}-\frac{2239}{5184}a^{5}-\frac{11}{24}a^{4}-\frac{205}{648}a^{3}-\frac{4}{27}a^{2}+\frac{67}{216}a-\frac{1}{3}$, $\frac{1}{41472}a^{26}+\frac{5}{41472}a^{22}-\frac{1}{20736}a^{20}+\frac{1}{1152}a^{18}+\frac{1}{1296}a^{17}-\frac{1}{864}a^{16}-\frac{1}{144}a^{15}+\frac{17}{3456}a^{14}-\frac{1}{1296}a^{13}+\frac{1}{576}a^{12}-\frac{13}{648}a^{11}+\frac{25}{1152}a^{10}+\frac{1}{72}a^{9}-\frac{97}{324}a^{8}-\frac{1}{216}a^{7}+\frac{205}{1152}a^{6}-\frac{61}{216}a^{5}+\frac{271}{5184}a^{4}+\frac{1}{36}a^{3}-\frac{121}{1296}a^{2}+\frac{5}{18}a+\frac{3}{8}$, $\frac{1}{22\cdots 16}a^{27}-\frac{14\cdots 35}{18\cdots 68}a^{26}-\frac{15\cdots 71}{11\cdots 08}a^{25}-\frac{10\cdots 11}{55\cdots 04}a^{24}-\frac{20\cdots 51}{24\cdots 24}a^{23}-\frac{32\cdots 41}{20\cdots 52}a^{22}+\frac{50\cdots 79}{27\cdots 52}a^{21}+\frac{11\cdots 85}{18\cdots 68}a^{20}-\frac{99\cdots 09}{18\cdots 68}a^{19}-\frac{22\cdots 71}{13\cdots 76}a^{18}+\frac{29\cdots 63}{91\cdots 84}a^{17}-\frac{45\cdots 09}{13\cdots 76}a^{16}-\frac{17\cdots 21}{55\cdots 04}a^{15}+\frac{27\cdots 29}{45\cdots 92}a^{14}-\frac{55\cdots 03}{10\cdots 56}a^{13}-\frac{99\cdots 55}{13\cdots 76}a^{12}+\frac{21\cdots 91}{18\cdots 68}a^{11}-\frac{18\cdots 97}{17\cdots 96}a^{10}-\frac{91\cdots 83}{27\cdots 52}a^{9}+\frac{88\cdots 29}{15\cdots 64}a^{8}+\frac{21\cdots 85}{55\cdots 04}a^{7}+\frac{14\cdots 05}{13\cdots 76}a^{6}+\frac{44\cdots 87}{22\cdots 96}a^{5}+\frac{16\cdots 55}{15\cdots 64}a^{4}+\frac{10\cdots 85}{68\cdots 88}a^{3}+\frac{19\cdots 95}{95\cdots 04}a^{2}-\frac{56\cdots 93}{14\cdots 06}a+\frac{23\cdots 13}{63\cdots 36}$ 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:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
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:   $\frac{24\cdots 93}{11\cdots 08}a^{27}+\frac{43\cdots 15}{11\cdots 08}a^{26}-\frac{60\cdots 43}{13\cdots 76}a^{25}+\frac{41\cdots 35}{55\cdots 04}a^{24}-\frac{63\cdots 09}{11\cdots 08}a^{23}+\frac{17\cdots 47}{11\cdots 08}a^{22}-\frac{84\cdots 73}{18\cdots 68}a^{21}+\frac{66\cdots 07}{34\cdots 44}a^{20}-\frac{19\cdots 45}{27\cdots 52}a^{19}+\frac{82\cdots 43}{27\cdots 52}a^{18}-\frac{64\cdots 83}{68\cdots 88}a^{17}+\frac{32\cdots 61}{13\cdots 76}a^{16}-\frac{22\cdots 39}{27\cdots 52}a^{15}+\frac{37\cdots 65}{27\cdots 52}a^{14}-\frac{62\cdots 13}{13\cdots 76}a^{13}+\frac{46\cdots 03}{11\cdots 48}a^{12}-\frac{33\cdots 65}{27\cdots 52}a^{11}+\frac{11\cdots 35}{27\cdots 52}a^{10}-\frac{17\cdots 83}{34\cdots 44}a^{9}-\frac{31\cdots 85}{13\cdots 76}a^{8}+\frac{47\cdots 67}{27\cdots 52}a^{7}+\frac{14\cdots 71}{27\cdots 52}a^{6}+\frac{70\cdots 57}{13\cdots 76}a^{5}-\frac{35\cdots 55}{17\cdots 72}a^{4}-\frac{59\cdots 37}{11\cdots 48}a^{3}-\frac{47\cdots 85}{34\cdots 44}a^{2}+\frac{23\cdots 45}{17\cdots 72}a+\frac{38\cdots 01}{47\cdots 02}$, $\frac{16\cdots 13}{13\cdots 76}a^{27}-\frac{12\cdots 81}{18\cdots 68}a^{26}+\frac{59\cdots 47}{27\cdots 52}a^{25}-\frac{26\cdots 76}{21\cdots 09}a^{24}+\frac{18\cdots 87}{68\cdots 88}a^{23}-\frac{27\cdots 99}{55\cdots 04}a^{22}+\frac{45\cdots 23}{27\cdots 52}a^{21}-\frac{71\cdots 11}{91\cdots 84}a^{20}+\frac{44\cdots 39}{17\cdots 72}a^{19}-\frac{16\cdots 01}{13\cdots 76}a^{18}+\frac{74\cdots 33}{22\cdots 96}a^{17}-\frac{27\cdots 59}{34\cdots 44}a^{16}+\frac{21\cdots 07}{68\cdots 88}a^{15}-\frac{38\cdots 21}{13\cdots 76}a^{14}+\frac{12\cdots 15}{68\cdots 88}a^{13}+\frac{23\cdots 01}{68\cdots 88}a^{12}+\frac{65\cdots 09}{11\cdots 48}a^{11}+\frac{71\cdots 21}{13\cdots 76}a^{10}+\frac{39\cdots 19}{68\cdots 88}a^{9}+\frac{11\cdots 43}{14\cdots 06}a^{8}-\frac{83\cdots 41}{34\cdots 44}a^{7}-\frac{10\cdots 43}{13\cdots 76}a^{6}-\frac{49\cdots 53}{68\cdots 88}a^{5}-\frac{37\cdots 09}{68\cdots 88}a^{4}-\frac{73\cdots 23}{43\cdots 18}a^{3}+\frac{36\cdots 11}{19\cdots 08}a^{2}+\frac{34\cdots 27}{86\cdots 36}a+\frac{14\cdots 43}{95\cdots 04}$, $\frac{14\cdots 89}{22\cdots 16}a^{27}+\frac{35\cdots 41}{68\cdots 88}a^{26}-\frac{13\cdots 23}{11\cdots 08}a^{25}+\frac{43\cdots 43}{45\cdots 92}a^{24}-\frac{11\cdots 95}{73\cdots 72}a^{23}+\frac{10\cdots 93}{34\cdots 44}a^{22}-\frac{54\cdots 09}{55\cdots 04}a^{21}+\frac{62\cdots 65}{13\cdots 76}a^{20}-\frac{85\cdots 21}{55\cdots 04}a^{19}+\frac{12\cdots 67}{17\cdots 72}a^{18}-\frac{54\cdots 47}{27\cdots 52}a^{17}+\frac{33\cdots 75}{68\cdots 88}a^{16}-\frac{33\cdots 81}{18\cdots 68}a^{15}+\frac{56\cdots 41}{28\cdots 12}a^{14}-\frac{14\cdots 69}{13\cdots 76}a^{13}+\frac{77\cdots 59}{34\cdots 44}a^{12}-\frac{17\cdots 77}{55\cdots 04}a^{11}-\frac{38\cdots 79}{17\cdots 72}a^{10}-\frac{75\cdots 11}{27\cdots 52}a^{9}-\frac{80\cdots 46}{21\cdots 09}a^{8}+\frac{11\cdots 91}{55\cdots 04}a^{7}+\frac{36\cdots 31}{95\cdots 04}a^{6}+\frac{49\cdots 03}{15\cdots 64}a^{5}+\frac{80\cdots 41}{34\cdots 44}a^{4}+\frac{30\cdots 79}{68\cdots 88}a^{3}-\frac{53\cdots 63}{43\cdots 18}a^{2}+\frac{74\cdots 69}{17\cdots 72}a-\frac{23\cdots 07}{47\cdots 02}$, $\frac{65\cdots 39}{55\cdots 04}a^{27}-\frac{95\cdots 39}{34\cdots 44}a^{26}+\frac{39\cdots 05}{18\cdots 68}a^{25}-\frac{45\cdots 79}{91\cdots 84}a^{24}+\frac{14\cdots 61}{55\cdots 04}a^{23}-\frac{62\cdots 49}{15\cdots 64}a^{22}+\frac{79\cdots 25}{55\cdots 04}a^{21}-\frac{65\cdots 39}{91\cdots 84}a^{20}+\frac{10\cdots 35}{45\cdots 92}a^{19}-\frac{38\cdots 23}{34\cdots 44}a^{18}+\frac{39\cdots 47}{13\cdots 76}a^{17}-\frac{74\cdots 31}{11\cdots 48}a^{16}+\frac{12\cdots 47}{45\cdots 92}a^{15}-\frac{54\cdots 75}{34\cdots 44}a^{14}+\frac{25\cdots 43}{15\cdots 64}a^{13}+\frac{74\cdots 07}{68\cdots 88}a^{12}+\frac{82\cdots 67}{15\cdots 64}a^{11}+\frac{51\cdots 58}{71\cdots 03}a^{10}+\frac{86\cdots 23}{13\cdots 76}a^{9}+\frac{59\cdots 73}{68\cdots 88}a^{8}-\frac{16\cdots 57}{45\cdots 92}a^{7}-\frac{36\cdots 13}{38\cdots 16}a^{6}-\frac{12\cdots 91}{13\cdots 76}a^{5}-\frac{50\cdots 61}{76\cdots 32}a^{4}-\frac{54\cdots 10}{21\cdots 09}a^{3}+\frac{76\cdots 89}{28\cdots 12}a^{2}+\frac{15\cdots 67}{57\cdots 24}a-\frac{41\cdots 19}{31\cdots 68}$, $\frac{15\cdots 99}{22\cdots 16}a^{27}-\frac{14\cdots 09}{11\cdots 08}a^{26}-\frac{13\cdots 13}{11\cdots 08}a^{25}-\frac{14\cdots 25}{61\cdots 56}a^{24}-\frac{34\cdots 35}{22\cdots 16}a^{23}+\frac{69\cdots 01}{40\cdots 04}a^{22}-\frac{42\cdots 33}{55\cdots 04}a^{21}+\frac{88\cdots 07}{22\cdots 96}a^{20}-\frac{65\cdots 87}{55\cdots 04}a^{19}+\frac{16\cdots 47}{27\cdots 52}a^{18}-\frac{38\cdots 25}{27\cdots 52}a^{17}+\frac{43\cdots 73}{13\cdots 76}a^{16}-\frac{27\cdots 51}{18\cdots 68}a^{15}+\frac{73\cdots 21}{27\cdots 52}a^{14}-\frac{44\cdots 15}{45\cdots 92}a^{13}-\frac{44\cdots 21}{43\cdots 18}a^{12}-\frac{64\cdots 77}{18\cdots 68}a^{11}-\frac{15\cdots 05}{27\cdots 52}a^{10}-\frac{15\cdots 01}{27\cdots 52}a^{9}-\frac{10\cdots 53}{13\cdots 76}a^{8}-\frac{11\cdots 83}{55\cdots 04}a^{7}+\frac{49\cdots 25}{91\cdots 84}a^{6}+\frac{10\cdots 87}{13\cdots 76}a^{5}+\frac{37\cdots 89}{57\cdots 24}a^{4}+\frac{25\cdots 37}{68\cdots 88}a^{3}+\frac{35\cdots 43}{38\cdots 16}a^{2}+\frac{17\cdots 37}{17\cdots 72}a+\frac{86\cdots 27}{47\cdots 02}$, $\frac{38\cdots 79}{27\cdots 52}a^{27}-\frac{19\cdots 67}{18\cdots 68}a^{26}+\frac{14\cdots 35}{55\cdots 04}a^{25}-\frac{18\cdots 81}{91\cdots 84}a^{24}+\frac{45\cdots 15}{13\cdots 76}a^{23}-\frac{36\cdots 01}{55\cdots 04}a^{22}+\frac{39\cdots 29}{18\cdots 68}a^{21}-\frac{14\cdots 53}{15\cdots 64}a^{20}+\frac{18\cdots 65}{57\cdots 24}a^{19}-\frac{20\cdots 69}{13\cdots 76}a^{18}+\frac{64\cdots 21}{15\cdots 64}a^{17}-\frac{70\cdots 59}{68\cdots 88}a^{16}+\frac{30\cdots 43}{76\cdots 32}a^{15}-\frac{57\cdots 75}{13\cdots 76}a^{14}+\frac{31\cdots 89}{13\cdots 76}a^{13}-\frac{11\cdots 59}{28\cdots 12}a^{12}+\frac{53\cdots 39}{76\cdots 32}a^{11}+\frac{75\cdots 51}{15\cdots 64}a^{10}+\frac{86\cdots 67}{13\cdots 76}a^{9}+\frac{19\cdots 43}{22\cdots 96}a^{8}-\frac{74\cdots 05}{17\cdots 72}a^{7}-\frac{35\cdots 07}{45\cdots 92}a^{6}-\frac{10\cdots 45}{13\cdots 76}a^{5}-\frac{20\cdots 87}{34\cdots 44}a^{4}-\frac{69\cdots 91}{63\cdots 36}a^{3}+\frac{63\cdots 31}{19\cdots 08}a^{2}-\frac{57\cdots 59}{57\cdots 24}a+\frac{20\cdots 87}{15\cdots 34}$, $\frac{12\cdots 27}{22\cdots 16}a^{27}-\frac{67\cdots 29}{18\cdots 68}a^{26}+\frac{38\cdots 47}{36\cdots 36}a^{25}-\frac{11\cdots 77}{17\cdots 72}a^{24}+\frac{32\cdots 31}{24\cdots 24}a^{23}-\frac{13\cdots 63}{55\cdots 04}a^{22}+\frac{44\cdots 23}{55\cdots 04}a^{21}-\frac{11\cdots 81}{30\cdots 28}a^{20}+\frac{23\cdots 21}{18\cdots 68}a^{19}-\frac{82\cdots 19}{13\cdots 76}a^{18}+\frac{14\cdots 23}{91\cdots 84}a^{17}-\frac{89\cdots 19}{22\cdots 96}a^{16}+\frac{83\cdots 57}{55\cdots 04}a^{15}-\frac{65\cdots 31}{45\cdots 92}a^{14}+\frac{12\cdots 71}{13\cdots 76}a^{13}+\frac{10\cdots 89}{68\cdots 88}a^{12}+\frac{48\cdots 37}{18\cdots 68}a^{11}+\frac{11\cdots 67}{45\cdots 92}a^{10}+\frac{62\cdots 21}{27\cdots 52}a^{9}+\frac{45\cdots 25}{12\cdots 72}a^{8}-\frac{35\cdots 43}{20\cdots 52}a^{7}-\frac{52\cdots 75}{13\cdots 76}a^{6}-\frac{13\cdots 07}{45\cdots 92}a^{5}-\frac{15\cdots 09}{68\cdots 88}a^{4}-\frac{29\cdots 69}{68\cdots 88}a^{3}+\frac{98\cdots 17}{57\cdots 24}a^{2}-\frac{29\cdots 45}{63\cdots 36}a+\frac{62\cdots 63}{10\cdots 56}$, $\frac{87\cdots 43}{25\cdots 44}a^{27}-\frac{17\cdots 91}{13\cdots 76}a^{26}-\frac{33\cdots 35}{55\cdots 04}a^{25}-\frac{21\cdots 29}{91\cdots 84}a^{24}-\frac{76\cdots 21}{10\cdots 76}a^{23}+\frac{29\cdots 11}{43\cdots 18}a^{22}-\frac{60\cdots 53}{18\cdots 68}a^{21}+\frac{16\cdots 07}{91\cdots 84}a^{20}-\frac{36\cdots 15}{68\cdots 88}a^{19}+\frac{21\cdots 39}{76\cdots 32}a^{18}-\frac{82\cdots 23}{13\cdots 76}a^{17}+\frac{44\cdots 67}{34\cdots 44}a^{16}-\frac{76\cdots 75}{11\cdots 48}a^{15}-\frac{10\cdots 15}{11\cdots 48}a^{14}-\frac{59\cdots 97}{13\cdots 76}a^{13}-\frac{14\cdots 39}{22\cdots 96}a^{12}-\frac{23\cdots 43}{14\cdots 06}a^{11}-\frac{52\cdots 47}{17\cdots 72}a^{10}-\frac{44\cdots 23}{17\cdots 96}a^{9}-\frac{23\cdots 09}{68\cdots 88}a^{8}-\frac{67\cdots 81}{68\cdots 88}a^{7}+\frac{19\cdots 27}{57\cdots 24}a^{6}+\frac{17\cdots 47}{45\cdots 92}a^{5}+\frac{20\cdots 01}{68\cdots 88}a^{4}+\frac{84\cdots 23}{57\cdots 24}a^{3}+\frac{23\cdots 57}{28\cdots 12}a^{2}-\frac{19\cdots 35}{17\cdots 72}a+\frac{16\cdots 53}{95\cdots 04}$, $\frac{89\cdots 87}{24\cdots 24}a^{27}+\frac{36\cdots 25}{11\cdots 08}a^{26}-\frac{24\cdots 93}{36\cdots 36}a^{25}+\frac{11\cdots 93}{18\cdots 68}a^{24}-\frac{62\cdots 57}{73\cdots 72}a^{23}+\frac{67\cdots 87}{36\cdots 36}a^{22}-\frac{15\cdots 25}{27\cdots 52}a^{21}+\frac{70\cdots 63}{27\cdots 52}a^{20}-\frac{53\cdots 03}{61\cdots 56}a^{19}+\frac{36\cdots 31}{91\cdots 84}a^{18}-\frac{31\cdots 63}{27\cdots 52}a^{17}+\frac{12\cdots 37}{45\cdots 92}a^{16}-\frac{18\cdots 31}{18\cdots 68}a^{15}+\frac{10\cdots 01}{91\cdots 84}a^{14}-\frac{66\cdots 25}{11\cdots 48}a^{13}+\frac{51\cdots 63}{68\cdots 88}a^{12}-\frac{88\cdots 23}{55\cdots 04}a^{11}-\frac{28\cdots 51}{30\cdots 28}a^{10}-\frac{75\cdots 25}{91\cdots 84}a^{9}-\frac{17\cdots 33}{13\cdots 76}a^{8}+\frac{37\cdots 43}{18\cdots 68}a^{7}+\frac{22\cdots 59}{91\cdots 84}a^{6}+\frac{33\cdots 67}{25\cdots 44}a^{5}-\frac{51\cdots 63}{22\cdots 96}a^{4}-\frac{66\cdots 67}{68\cdots 88}a^{3}-\frac{59\cdots 15}{34\cdots 44}a^{2}+\frac{24\cdots 70}{26\cdots 89}a-\frac{37\cdots 19}{10\cdots 56}$, $\frac{75\cdots 77}{73\cdots 72}a^{27}-\frac{30\cdots 11}{36\cdots 36}a^{26}+\frac{21\cdots 09}{11\cdots 08}a^{25}-\frac{85\cdots 13}{55\cdots 04}a^{24}+\frac{18\cdots 29}{73\cdots 72}a^{23}-\frac{18\cdots 71}{36\cdots 36}a^{22}+\frac{87\cdots 17}{55\cdots 04}a^{21}-\frac{49\cdots 91}{68\cdots 88}a^{20}+\frac{13\cdots 95}{55\cdots 04}a^{19}-\frac{10\cdots 67}{91\cdots 84}a^{18}+\frac{28\cdots 83}{91\cdots 84}a^{17}-\frac{10\cdots 39}{13\cdots 76}a^{16}+\frac{16\cdots 69}{55\cdots 04}a^{15}-\frac{29\cdots 49}{91\cdots 84}a^{14}+\frac{26\cdots 69}{15\cdots 64}a^{13}-\frac{35\cdots 05}{34\cdots 44}a^{12}+\frac{27\cdots 39}{55\cdots 04}a^{11}+\frac{91\cdots 31}{27\cdots 52}a^{10}+\frac{13\cdots 45}{30\cdots 28}a^{9}+\frac{86\cdots 11}{15\cdots 64}a^{8}-\frac{18\cdots 45}{55\cdots 04}a^{7}-\frac{16\cdots 97}{27\cdots 52}a^{6}-\frac{22\cdots 05}{45\cdots 92}a^{5}-\frac{25\cdots 48}{71\cdots 03}a^{4}-\frac{43\cdots 85}{68\cdots 88}a^{3}+\frac{69\cdots 87}{34\cdots 44}a^{2}-\frac{85\cdots 25}{17\cdots 72}a+\frac{37\cdots 13}{47\cdots 02}$, $\frac{10\cdots 99}{40\cdots 04}a^{27}+\frac{10\cdots 97}{55\cdots 04}a^{26}-\frac{25\cdots 71}{55\cdots 04}a^{25}+\frac{23\cdots 21}{68\cdots 88}a^{24}-\frac{64\cdots 85}{11\cdots 08}a^{23}+\frac{63\cdots 69}{55\cdots 04}a^{22}-\frac{34\cdots 07}{91\cdots 84}a^{21}+\frac{46\cdots 03}{27\cdots 52}a^{20}-\frac{15\cdots 49}{27\cdots 52}a^{19}+\frac{12\cdots 81}{45\cdots 92}a^{18}-\frac{10\cdots 27}{13\cdots 76}a^{17}+\frac{61\cdots 57}{34\cdots 44}a^{16}-\frac{18\cdots 79}{27\cdots 52}a^{15}+\frac{99\cdots 91}{13\cdots 76}a^{14}-\frac{27\cdots 89}{68\cdots 88}a^{13}+\frac{18\cdots 13}{22\cdots 96}a^{12}-\frac{32\cdots 25}{27\cdots 52}a^{11}-\frac{11\cdots 19}{13\cdots 76}a^{10}-\frac{46\cdots 45}{45\cdots 92}a^{9}-\frac{24\cdots 89}{17\cdots 72}a^{8}+\frac{21\cdots 59}{27\cdots 52}a^{7}+\frac{19\cdots 13}{13\cdots 76}a^{6}+\frac{83\cdots 23}{68\cdots 88}a^{5}+\frac{61\cdots 07}{68\cdots 88}a^{4}+\frac{66\cdots 23}{38\cdots 16}a^{3}-\frac{82\cdots 29}{17\cdots 72}a^{2}+\frac{14\cdots 93}{86\cdots 36}a-\frac{17\cdots 89}{95\cdots 04}$, $\frac{89\cdots 75}{13\cdots 76}a^{27}-\frac{15\cdots 45}{27\cdots 52}a^{26}+\frac{11\cdots 31}{91\cdots 84}a^{25}-\frac{32\cdots 51}{30\cdots 28}a^{24}+\frac{10\cdots 11}{68\cdots 88}a^{23}-\frac{88\cdots 75}{27\cdots 52}a^{22}+\frac{28\cdots 37}{27\cdots 52}a^{21}-\frac{42\cdots 47}{91\cdots 84}a^{20}+\frac{54\cdots 67}{34\cdots 44}a^{19}-\frac{12\cdots 23}{17\cdots 72}a^{18}+\frac{14\cdots 61}{68\cdots 88}a^{17}-\frac{57\cdots 11}{11\cdots 48}a^{16}+\frac{43\cdots 33}{22\cdots 96}a^{15}-\frac{14\cdots 01}{68\cdots 88}a^{14}+\frac{75\cdots 35}{68\cdots 88}a^{13}-\frac{95\cdots 41}{68\cdots 88}a^{12}+\frac{37\cdots 07}{11\cdots 48}a^{11}+\frac{13\cdots 01}{68\cdots 88}a^{10}+\frac{18\cdots 45}{68\cdots 88}a^{9}+\frac{23\cdots 99}{68\cdots 88}a^{8}-\frac{90\cdots 77}{38\cdots 16}a^{7}-\frac{79\cdots 69}{22\cdots 96}a^{6}-\frac{20\cdots 31}{68\cdots 88}a^{5}-\frac{14\cdots 21}{68\cdots 88}a^{4}-\frac{26\cdots 21}{86\cdots 36}a^{3}+\frac{13\cdots 47}{14\cdots 06}a^{2}-\frac{17\cdots 11}{86\cdots 36}a+\frac{37\cdots 11}{95\cdots 04}$, $\frac{10\cdots 73}{73\cdots 72}a^{27}+\frac{22\cdots 35}{68\cdots 88}a^{26}+\frac{27\cdots 39}{11\cdots 08}a^{25}+\frac{13\cdots 19}{22\cdots 96}a^{24}+\frac{23\cdots 33}{73\cdots 72}a^{23}-\frac{75\cdots 13}{22\cdots 96}a^{22}+\frac{42\cdots 95}{27\cdots 52}a^{21}-\frac{53\cdots 95}{68\cdots 88}a^{20}+\frac{43\cdots 01}{18\cdots 68}a^{19}-\frac{13\cdots 23}{11\cdots 48}a^{18}+\frac{76\cdots 79}{27\cdots 52}a^{17}-\frac{43\cdots 69}{68\cdots 88}a^{16}+\frac{54\cdots 51}{18\cdots 68}a^{15}-\frac{53\cdots 45}{11\cdots 48}a^{14}+\frac{22\cdots 37}{11\cdots 48}a^{13}+\frac{92\cdots 91}{43\cdots 18}a^{12}+\frac{39\cdots 23}{55\cdots 04}a^{11}+\frac{37\cdots 13}{31\cdots 68}a^{10}+\frac{11\cdots 81}{91\cdots 84}a^{9}+\frac{52\cdots 15}{34\cdots 44}a^{8}+\frac{27\cdots 99}{55\cdots 04}a^{7}-\frac{10\cdots 43}{95\cdots 04}a^{6}-\frac{11\cdots 83}{76\cdots 32}a^{5}-\frac{26\cdots 49}{19\cdots 08}a^{4}-\frac{54\cdots 69}{68\cdots 88}a^{3}-\frac{96\cdots 49}{43\cdots 18}a^{2}-\frac{33\cdots 93}{14\cdots 06}a-\frac{36\cdots 41}{79\cdots 67}$, $\frac{18\cdots 55}{11\cdots 08}a^{27}+\frac{31\cdots 93}{11\cdots 08}a^{26}-\frac{18\cdots 83}{55\cdots 04}a^{25}+\frac{29\cdots 17}{55\cdots 04}a^{24}-\frac{48\cdots 83}{11\cdots 08}a^{23}+\frac{12\cdots 81}{11\cdots 08}a^{22}-\frac{11\cdots 25}{34\cdots 44}a^{21}+\frac{61\cdots 27}{42\cdots 24}a^{20}-\frac{14\cdots 51}{27\cdots 52}a^{19}+\frac{61\cdots 93}{27\cdots 52}a^{18}-\frac{95\cdots 91}{13\cdots 76}a^{17}+\frac{24\cdots 45}{13\cdots 76}a^{16}-\frac{16\cdots 01}{27\cdots 52}a^{15}+\frac{27\cdots 87}{27\cdots 52}a^{14}-\frac{29\cdots 77}{86\cdots 36}a^{13}+\frac{48\cdots 99}{17\cdots 72}a^{12}-\frac{88\cdots 73}{91\cdots 84}a^{11}+\frac{53\cdots 05}{27\cdots 52}a^{10}-\frac{80\cdots 59}{13\cdots 76}a^{9}-\frac{69\cdots 79}{13\cdots 76}a^{8}+\frac{31\cdots 77}{27\cdots 52}a^{7}+\frac{29\cdots 21}{27\cdots 52}a^{6}+\frac{31\cdots 79}{86\cdots 36}a^{5}+\frac{94\cdots 35}{43\cdots 18}a^{4}-\frac{48\cdots 07}{34\cdots 44}a^{3}+\frac{45\cdots 55}{11\cdots 48}a^{2}+\frac{81\cdots 01}{43\cdots 18}a+\frac{53\cdots 09}{47\cdots 02}$, $\frac{42\cdots 63}{22\cdots 16}a^{27}-\frac{44\cdots 97}{30\cdots 28}a^{26}-\frac{12\cdots 73}{36\cdots 36}a^{25}-\frac{38\cdots 65}{13\cdots 76}a^{24}-\frac{89\cdots 75}{22\cdots 16}a^{23}+\frac{14\cdots 39}{27\cdots 52}a^{22}-\frac{32\cdots 37}{17\cdots 72}a^{21}+\frac{24\cdots 99}{22\cdots 96}a^{20}-\frac{59\cdots 57}{18\cdots 68}a^{19}+\frac{11\cdots 75}{68\cdots 88}a^{18}-\frac{34\cdots 33}{91\cdots 84}a^{17}+\frac{46\cdots 87}{57\cdots 24}a^{16}-\frac{21\cdots 77}{55\cdots 04}a^{15}+\frac{37\cdots 83}{68\cdots 88}a^{14}-\frac{16\cdots 45}{68\cdots 88}a^{13}-\frac{92\cdots 19}{34\cdots 44}a^{12}-\frac{13\cdots 57}{18\cdots 68}a^{11}-\frac{11\cdots 21}{85\cdots 48}a^{10}-\frac{23\cdots 83}{27\cdots 52}a^{9}-\frac{10\cdots 13}{11\cdots 48}a^{8}+\frac{38\cdots 99}{18\cdots 68}a^{7}+\frac{16\cdots 43}{68\cdots 88}a^{6}+\frac{58\cdots 85}{34\cdots 44}a^{5}+\frac{20\cdots 95}{86\cdots 36}a^{4}-\frac{16\cdots 95}{68\cdots 88}a^{3}-\frac{40\cdots 37}{28\cdots 12}a^{2}+\frac{32\cdots 07}{28\cdots 12}a-\frac{70\cdots 86}{79\cdots 67}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 10821496099847764000 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{12}\cdot 10821496099847764000 \cdot 1}{2\cdot\sqrt{4717667136961114442493020206737586382842235909528214306816}}\cr\approx \mathstrut & 4.77168830360644 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600) 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^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600, 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^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 18*x^26 + 225*x^24 - 288*x^23 + 1152*x^22 - 5760*x^21 + 18108*x^20 - 89952*x^19 + 217224*x^18 - 499680*x^17 + 2230524*x^16 - 814464*x^15 + 13974624*x^14 + 12253824*x^13 + 48314916*x^12 + 71792064*x^11 + 68082600*x^10 + 88956864*x^9 + 11560068*x^8 - 83134080*x^7 - 93788352*x^6 - 73624320*x^5 - 34564032*x^4 - 3382272*x^3 + 1400832*x^2 + 22528*x + 57600); 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

$G(2,2)$ (as 28T393):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 12096
The 16 conjugacy class representatives for $G(2,2)$
Character table for $G(2,2)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 36 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.6.0.1}{6} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.8.0.1}{8} }^{3}{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.7.0.1}{7} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.12.0.1}{12} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }^{4}$ ${\href{/padicField/41.7.0.1}{7} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{3}{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.6.0.1}{6} }^{4}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.8.0.1}{8} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.1.4.9a1.1$x^{4} + 2 x^{2} + 2$$4$$1$$9$$D_{4}$$$[2, 3, \frac{7}{2}]$$
2.1.8.31a1.155$x^{8} + 4 x^{4} + 18$$8$$1$$31$$(C_8:C_2):C_2$$$[2, 3, \frac{7}{2}, 4, 5]$$
2.1.16.66h1.325$x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{10} + 16 x^{9} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 18$$16$$1$$66$16T41$$[2, 3, \frac{7}{2}, 4, 5]$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
Deg $27$$27$$1$$54$

Spectrum of ring of integers

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