Properties

Label 40.0.287...000.1
Degree $40$
Signature $(0, 20)$
Discriminant $2.879\times 10^{98}$
Root discriminant \(289.39\)
Ramified primes $2,5,11$
Class number not computed
Class group not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025)
 
Copy content gp:K = bnfinit(y^40 + 990*y^36 + 341825*y^32 - 128700*y^30 + 45465750*y^28 - 95559750*y^26 - 187005500*y^24 - 13476375000*y^22 - 826202149740*y^20 + 2063996673750*y^18 - 40426833250925*y^16 + 260603669081250*y^14 + 3310939947063375*y^12 + 8689693619326500*y^10 + 363555244286551625*y^8 - 62474494850460000*y^6 + 9400451839944969375*y^4 + 765165773393381250*y^2 + 677576408158587025, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025)
 

\( x^{40} + 990 x^{36} + 341825 x^{32} - 128700 x^{30} + 45465750 x^{28} - 95559750 x^{26} + \cdots + 67\!\cdots\!25 \) 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:  $40$
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:  $(0, 20)$
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:   \(287\!\cdots\!000\) \(\medspace = 2^{40}\cdot 5^{70}\cdot 11^{36}\) 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:  \(289.39\)
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:  $2\cdot 5^{7/4}11^{9/10}\approx 289.3882675684651$
Ramified primes:   \(2\), \(5\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2\times C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1100=2^{2}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1100}(1,·)$, $\chi_{1100}(261,·)$, $\chi_{1100}(647,·)$, $\chi_{1100}(269,·)$, $\chi_{1100}(1041,·)$, $\chi_{1100}(789,·)$, $\chi_{1100}(1047,·)$, $\chi_{1100}(281,·)$, $\chi_{1100}(901,·)$, $\chi_{1100}(289,·)$, $\chi_{1100}(243,·)$, $\chi_{1100}(549,·)$, $\chi_{1100}(1063,·)$, $\chi_{1100}(43,·)$, $\chi_{1100}(307,·)$, $\chi_{1100}(181,·)$, $\chi_{1100}(567,·)$, $\chi_{1100}(1083,·)$, $\chi_{1100}(321,·)$, $\chi_{1100}(927,·)$, $\chi_{1100}(327,·)$, $\chi_{1100}(587,·)$, $\chi_{1100}(909,·)$, $\chi_{1100}(467,·)$, $\chi_{1100}(723,·)$, $\chi_{1100}(603,·)$, $\chi_{1100}(861,·)$, $\chi_{1100}(223,·)$, $\chi_{1100}(609,·)$, $\chi_{1100}(229,·)$, $\chi_{1100}(741,·)$, $\chi_{1100}(1003,·)$, $\chi_{1100}(749,·)$, $\chi_{1100}(369,·)$, $\chi_{1100}(83,·)$, $\chi_{1100}(629,·)$, $\chi_{1100}(887,·)$, $\chi_{1100}(763,·)$, $\chi_{1100}(1021,·)$, $\chi_{1100}(507,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{33}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{33}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{33}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{33}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{33}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{33}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{99}a^{16}+\frac{1}{9}a^{8}-\frac{4}{9}$, $\frac{1}{1089}a^{17}-\frac{1}{9}a^{9}-\frac{1}{11}a^{7}+\frac{1}{9}a$, $\frac{1}{1089}a^{18}+\frac{1}{99}a^{10}-\frac{1}{11}a^{8}-\frac{2}{9}a^{2}$, $\frac{1}{1089}a^{19}+\frac{1}{99}a^{11}-\frac{1}{11}a^{9}-\frac{2}{9}a^{3}$, $\frac{1}{21780}a^{20}-\frac{1}{2178}a^{18}-\frac{1}{396}a^{16}-\frac{1}{66}a^{14}-\frac{1}{99}a^{12}+\frac{1}{99}a^{10}-\frac{59}{396}a^{8}+\frac{5}{12}a^{6}-\frac{7}{36}a^{4}+\frac{7}{36}a^{2}-\frac{11}{36}$, $\frac{1}{21780}a^{21}-\frac{1}{2178}a^{19}+\frac{1}{4356}a^{17}-\frac{1}{66}a^{15}-\frac{1}{99}a^{13}+\frac{1}{99}a^{11}-\frac{59}{396}a^{9}+\frac{19}{132}a^{7}-\frac{7}{36}a^{5}+\frac{7}{36}a^{3}+\frac{13}{36}a$, $\frac{1}{21780}a^{22}+\frac{1}{4356}a^{18}-\frac{1}{99}a^{14}+\frac{1}{396}a^{10}-\frac{1}{44}a^{8}+\frac{11}{36}a^{6}+\frac{1}{4}a^{4}+\frac{7}{36}a^{2}-\frac{1}{2}$, $\frac{1}{21780}a^{23}+\frac{1}{4356}a^{19}-\frac{1}{99}a^{15}+\frac{1}{396}a^{11}-\frac{1}{44}a^{9}+\frac{11}{36}a^{7}+\frac{1}{4}a^{5}+\frac{7}{36}a^{3}-\frac{1}{2}a$, $\frac{1}{65340}a^{24}-\frac{1}{2178}a^{18}-\frac{1}{396}a^{16}-\frac{1}{66}a^{14}+\frac{1}{132}a^{12}+\frac{1}{396}a^{10}-\frac{4}{33}a^{8}+\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{36}a^{2}-\frac{25}{108}$, $\frac{1}{718740}a^{25}-\frac{1}{2178}a^{19}-\frac{1}{4356}a^{17}-\frac{3}{242}a^{15}-\frac{1}{132}a^{13}-\frac{5}{396}a^{11}-\frac{4}{33}a^{9}-\frac{5}{22}a^{7}+\frac{13}{66}a^{5}+\frac{7}{36}a^{3}+\frac{1}{108}a$, $\frac{1}{718740}a^{26}-\frac{1}{4356}a^{18}+\frac{1}{363}a^{16}-\frac{1}{132}a^{14}+\frac{1}{132}a^{12}-\frac{2}{33}a^{8}-\frac{10}{33}a^{6}-\frac{1}{12}a^{4}+\frac{19}{108}a^{2}-\frac{1}{6}$, $\frac{1}{718740}a^{27}-\frac{1}{4356}a^{19}-\frac{1}{132}a^{15}+\frac{1}{132}a^{13}-\frac{2}{33}a^{9}-\frac{1}{33}a^{7}-\frac{1}{12}a^{5}+\frac{19}{108}a^{3}+\frac{1}{6}a$, $\frac{1}{2156220}a^{28}-\frac{1}{2156220}a^{26}+\frac{1}{196020}a^{24}+\frac{1}{65340}a^{22}+\frac{1}{6534}a^{18}-\frac{23}{13068}a^{16}-\frac{5}{594}a^{14}+\frac{4}{297}a^{12}+\frac{5}{594}a^{10}-\frac{8}{99}a^{8}-\frac{353}{1188}a^{6}-\frac{8}{81}a^{4}-\frac{40}{81}a^{2}+\frac{37}{162}$, $\frac{1}{2156220}a^{29}-\frac{1}{2156220}a^{27}-\frac{1}{2156220}a^{25}+\frac{1}{65340}a^{23}+\frac{1}{6534}a^{19}+\frac{1}{13068}a^{17}+\frac{71}{6534}a^{15}+\frac{4}{297}a^{13}+\frac{5}{594}a^{11}+\frac{14}{99}a^{9}-\frac{173}{1188}a^{7}+\frac{398}{891}a^{5}-\frac{40}{81}a^{3}-\frac{5}{162}a$, $\frac{1}{118592100}a^{30}+\frac{1}{2156220}a^{26}+\frac{1}{196020}a^{24}+\frac{1}{65340}a^{22}-\frac{1}{71874}a^{20}-\frac{1}{2178}a^{18}+\frac{1}{6534}a^{16}+\frac{5}{396}a^{14}+\frac{7}{594}a^{12}-\frac{593}{65340}a^{10}-\frac{7}{108}a^{8}+\frac{1343}{3564}a^{6}-\frac{13}{108}a^{4}+\frac{28}{81}a^{2}+\frac{131}{324}$, $\frac{1}{118592100}a^{31}+\frac{1}{2156220}a^{27}-\frac{1}{2156220}a^{25}+\frac{1}{65340}a^{23}-\frac{1}{71874}a^{21}-\frac{1}{2178}a^{19}+\frac{1}{6534}a^{17}+\frac{7}{4356}a^{15}+\frac{7}{594}a^{13}-\frac{593}{65340}a^{11}+\frac{5}{108}a^{9}+\frac{155}{3564}a^{7}+\frac{505}{1188}a^{5}+\frac{28}{81}a^{3}-\frac{133}{324}a$, $\frac{1}{355776300}a^{32}+\frac{1}{2156220}a^{26}+\frac{1}{294030}a^{24}+\frac{1}{179685}a^{22}-\frac{1}{3267}a^{18}+\frac{7}{19602}a^{16}+\frac{13}{1188}a^{14}+\frac{19}{10890}a^{12}+\frac{365}{2673}a^{8}+\frac{353}{1188}a^{6}+\frac{4}{9}a^{4}-\frac{53}{324}a^{2}+\frac{13}{243}$, $\frac{1}{63683957700}a^{33}+\frac{1}{321636150}a^{31}+\frac{1}{38596338}a^{29}+\frac{1}{64327230}a^{27}-\frac{61}{231578028}a^{25}-\frac{17}{3573735}a^{23}+\frac{71}{3898620}a^{21}-\frac{17}{2339172}a^{19}+\frac{31}{3508758}a^{17}+\frac{488}{64977}a^{15}-\frac{15674}{2923965}a^{13}-\frac{7033}{1063260}a^{11}-\frac{180925}{1913868}a^{9}-\frac{23467}{53163}a^{7}-\frac{109553}{318978}a^{5}+\frac{1349}{3222}a^{3}+\frac{36569}{86994}a$, $\frac{1}{63683957700}a^{34}+\frac{19}{63683957700}a^{32}+\frac{13}{21227985900}a^{30}+\frac{1}{64327230}a^{28}+\frac{769}{1157890140}a^{26}+\frac{1655}{231578028}a^{24}+\frac{1627}{128654460}a^{22}-\frac{368}{32163615}a^{20}+\frac{599}{7017516}a^{18}-\frac{2965}{7017516}a^{16}-\frac{21503}{5847930}a^{14}-\frac{38699}{11695860}a^{12}+\frac{1155359}{105262740}a^{10}+\frac{48935}{1913868}a^{8}-\frac{49409}{318978}a^{6}-\frac{115}{2148}a^{4}+\frac{30715}{173988}a^{2}-\frac{71}{486}$, $\frac{1}{484061762477700}a^{35}+\frac{4}{733426912845}a^{33}-\frac{4451}{14668538256900}a^{31}+\frac{5443}{88900231860}a^{29}-\frac{286207}{800102086740}a^{27}+\frac{4259}{488951275230}a^{25}-\frac{751769}{88900231860}a^{23}+\frac{126469}{7408352655}a^{21}+\frac{95653}{220413798}a^{19}+\frac{345487}{808183926}a^{17}+\frac{402373789}{88900231860}a^{15}-\frac{6674747}{538789284}a^{13}-\frac{74184889}{36368276670}a^{11}-\frac{35981629}{220413798}a^{9}+\frac{82563167}{440827596}a^{7}+\frac{8160349}{73471266}a^{5}+\frac{46111085}{120225708}a^{3}-\frac{19021}{742134}a$, $\frac{1}{38\cdots 00}a^{36}+\frac{15024781351}{70\cdots 80}a^{34}+\frac{4331940173483}{88\cdots 25}a^{32}-\frac{25367250278041}{11\cdots 00}a^{30}-\frac{27479791722167}{12\cdots 16}a^{28}-\frac{542542933921201}{35\cdots 90}a^{26}+\frac{852677102982443}{16\cdots 95}a^{24}+\frac{18317036116534}{17\cdots 55}a^{22}+\frac{322741947396547}{21\cdots 60}a^{20}+\frac{681539221706155}{39\cdots 52}a^{18}+\frac{11\cdots 41}{10\cdots 30}a^{16}-\frac{13\cdots 63}{21\cdots 14}a^{14}-\frac{14\cdots 72}{13\cdots 95}a^{12}+\frac{33\cdots 07}{58\cdots 80}a^{10}-\frac{83\cdots 15}{53\cdots 98}a^{8}+\frac{89\cdots 61}{17\cdots 66}a^{6}-\frac{58\cdots 49}{48\cdots 18}a^{4}-\frac{43\cdots 77}{96\cdots 36}a^{2}-\frac{238347695093}{782143970724}$, $\frac{1}{38\cdots 00}a^{37}-\frac{9528526}{97\cdots 75}a^{35}+\frac{42898010899}{59\cdots 50}a^{33}-\frac{9627809879521}{11\cdots 00}a^{31}+\frac{12771417052661}{16\cdots 95}a^{29}-\frac{88525613007109}{70\cdots 38}a^{27}-\frac{93454384072253}{19\cdots 05}a^{25}+\frac{327057907786562}{17\cdots 55}a^{23}+\frac{166441900281211}{53\cdots 65}a^{21}-\frac{274707421941433}{39\cdots 52}a^{19}+\frac{13\cdots 77}{35\cdots 10}a^{17}+\frac{78\cdots 01}{17\cdots 55}a^{15}+\frac{63\cdots 61}{14\cdots 45}a^{13}-\frac{22\cdots 33}{58\cdots 80}a^{11}+\frac{92\cdots 03}{88\cdots 33}a^{9}-\frac{38\cdots 53}{35\cdots 32}a^{7}+\frac{45\cdots 64}{26\cdots 99}a^{5}+\frac{767389579860244}{24\cdots 09}a^{3}-\frac{124417228173067}{298588288873089}a$, $\frac{1}{27\cdots 00}a^{38}+\frac{15\cdots 63}{13\cdots 50}a^{36}-\frac{17\cdots 61}{24\cdots 00}a^{34}-\frac{13\cdots 73}{24\cdots 00}a^{32}+\frac{25\cdots 31}{61\cdots 25}a^{30}-\frac{25\cdots 63}{12\cdots 05}a^{28}-\frac{27\cdots 59}{98\cdots 04}a^{26}-\frac{68\cdots 97}{22\cdots 10}a^{24}+\frac{12\cdots 17}{14\cdots 40}a^{22}-\frac{39\cdots 39}{29\cdots 88}a^{20}-\frac{68\cdots 77}{37\cdots 85}a^{18}+\frac{14\cdots 07}{14\cdots 40}a^{16}+\frac{10\cdots 27}{20\cdots 10}a^{14}-\frac{23\cdots 09}{20\cdots 10}a^{12}-\frac{59\cdots 71}{40\cdots 20}a^{10}+\frac{26\cdots 29}{37\cdots 42}a^{8}+\frac{21\cdots 09}{74\cdots 84}a^{6}-\frac{17\cdots 13}{67\cdots 44}a^{4}-\frac{17\cdots 09}{67\cdots 44}a^{2}-\frac{10\cdots 45}{54\cdots 96}$, $\frac{1}{27\cdots 00}a^{39}+\frac{15\cdots 63}{13\cdots 50}a^{37}-\frac{34\cdots 63}{27\cdots 00}a^{35}+\frac{13\cdots 57}{50\cdots 50}a^{33}-\frac{91\cdots 17}{24\cdots 00}a^{31}-\frac{62\cdots 27}{49\cdots 20}a^{29}-\frac{32\cdots 33}{24\cdots 10}a^{27}+\frac{50\cdots 13}{54\cdots 80}a^{25}+\frac{25\cdots 49}{14\cdots 40}a^{23}+\frac{89\cdots 27}{14\cdots 40}a^{21}+\frac{45\cdots 81}{74\cdots 70}a^{19}+\frac{39\cdots 39}{49\cdots 80}a^{17}-\frac{27\cdots 79}{22\cdots 10}a^{15}-\frac{38\cdots 43}{40\cdots 20}a^{13}+\frac{34\cdots 23}{40\cdots 20}a^{11}-\frac{68\cdots 53}{82\cdots 76}a^{9}-\frac{74\cdots 85}{18\cdots 71}a^{7}+\frac{19\cdots 50}{18\cdots 71}a^{5}-\frac{31\cdots 37}{67\cdots 44}a^{3}-\frac{17\cdots 41}{75\cdots 16}a$ 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:  Not computed
Index:  $1$
Inessential primes:  None

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:  $19$
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot R \cdot h}{2\cdot\sqrt{287896772221388971765277468942920243004336953163146972656250000000000000000000000000000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025) 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^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025, 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^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025); 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^40 + 990*x^36 + 341825*x^32 - 128700*x^30 + 45465750*x^28 - 95559750*x^26 - 187005500*x^24 - 13476375000*x^22 - 826202149740*x^20 + 2063996673750*x^18 - 40426833250925*x^16 + 260603669081250*x^14 + 3310939947063375*x^12 + 8689693619326500*x^10 + 363555244286551625*x^8 - 62474494850460000*x^6 + 9400451839944969375*x^4 + 765165773393381250*x^2 + 677576408158587025); 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

$C_2\times C_{20}$ (as 40T2):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{-110 +22 \sqrt{5}})\), 5.5.5719140625.2, 8.0.58564000000.1, 10.0.359794264373779296875.3, 10.10.163542847442626953125.1, 10.0.1798971321868896484375.1, 20.0.3236297816906724754371680319309234619140625.4, 20.20.140227447093420901489257812500000000000000000000.1, 20.0.16967521098303929080200195312500000000000000000000.3

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{10}$ R $20^{2}$ R ${\href{/padicField/13.4.0.1}{4} }^{10}$ $20^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{10}$ ${\href{/padicField/47.4.0.1}{4} }^{10}$ $20^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{8}$

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 2.10.2.20a1.2$x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 10 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$$2$$10$$20$20T1$$[2]^{10}$$
2.10.2.20a1.2$x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 10 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$$2$$10$$20$20T1$$[2]^{10}$$
\(5\) Copy content Toggle raw display 5.1.20.35a1.4$x^{20} + 20 x^{16} + 80$$20$$1$$35$20T1$$[2]_{4}$$
5.1.20.35a1.4$x^{20} + 20 x^{16} + 80$$20$$1$$35$20T1$$[2]_{4}$$
\(11\) Copy content Toggle raw display 11.2.10.18a1.10$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35884 x + 1123$$10$$2$$18$20T3$$[\ ]_{10}^{2}$$
11.2.10.18a1.10$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35884 x + 1123$$10$$2$$18$20T3$$[\ ]_{10}^{2}$$

Spectrum of ring of integers

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