↑

Properties

Label 20.0.359...449.1
Degree $20$
Signature $(0, 10)$
Discriminant $3.591\times 10^{28}$
Root discriminant \(26.78\)
Ramified primes $3,239$
Class number $3$
Class group [3]
Galois group $D_{10}$ (as 20T4)

Related objects

Downloads

Learn more

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^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49)
 
Copy content gp:K = bnfinit(y^20 - 10*y^19 + 39*y^18 - 66*y^17 - 16*y^16 + 332*y^15 - 576*y^14 - 554*y^13 + 3490*y^12 - 4378*y^11 - 109*y^10 + 4411*y^9 - 1459*y^8 - 5238*y^7 + 6204*y^6 - 605*y^5 - 1802*y^4 - 524*y^3 + 1329*y^2 - 469*y + 49, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49)
 

\( x^{20} - 10 x^{19} + 39 x^{18} - 66 x^{17} - 16 x^{16} + 332 x^{15} - 576 x^{14} - 554 x^{13} + \cdots + 49 \) 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:  $20$
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:  $(0, 10)$
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:   \(35908028125401873392383429449\) \(\medspace = 3^{10}\cdot 239^{10}\) 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:  \(26.78\)
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:  $3^{1/2}239^{1/2}\approx 26.77685567799177$
Ramified primes:   \(3\), \(239\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_{10}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-3}, \sqrt{-239})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}+\frac{3}{7}a^{3}-\frac{3}{7}a$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{8}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{9}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a$, $\frac{1}{7}a^{14}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{7}a^{15}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{2}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}$, $\frac{1}{49}a^{16}-\frac{1}{49}a^{15}+\frac{1}{49}a^{14}+\frac{1}{49}a^{10}+\frac{1}{49}a^{9}+\frac{12}{49}a^{8}+\frac{13}{49}a^{7}-\frac{22}{49}a^{6}+\frac{10}{49}a^{5}+\frac{19}{49}a^{4}+\frac{23}{49}a^{3}-\frac{9}{49}a^{2}$, $\frac{1}{49}a^{17}+\frac{1}{49}a^{14}+\frac{1}{49}a^{11}+\frac{2}{49}a^{10}+\frac{13}{49}a^{9}-\frac{24}{49}a^{8}-\frac{9}{49}a^{7}-\frac{12}{49}a^{6}-\frac{20}{49}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}-\frac{9}{49}a^{2}$, $\frac{1}{61080815789}a^{18}-\frac{9}{61080815789}a^{17}+\frac{349986771}{61080815789}a^{16}-\frac{399984852}{8725830827}a^{15}+\frac{2780963548}{61080815789}a^{14}+\frac{479129477}{8725830827}a^{13}+\frac{638827743}{61080815789}a^{12}-\frac{44615005}{8725830827}a^{11}+\frac{484391659}{61080815789}a^{10}+\frac{3467978869}{8725830827}a^{9}-\frac{153888099}{1001324849}a^{8}+\frac{9721018662}{61080815789}a^{7}+\frac{20349816413}{61080815789}a^{6}-\frac{1998657940}{61080815789}a^{5}+\frac{20075211574}{61080815789}a^{4}+\frac{2331754683}{8725830827}a^{3}+\frac{20407522641}{61080815789}a^{2}+\frac{2557126050}{8725830827}a+\frac{216459101}{1246547261}$, $\frac{1}{167422516077649}a^{19}+\frac{1361}{167422516077649}a^{18}-\frac{902150242523}{167422516077649}a^{17}+\frac{1314361741698}{167422516077649}a^{16}+\frac{983584849372}{167422516077649}a^{15}+\frac{2212707283975}{167422516077649}a^{14}+\frac{2867776008427}{167422516077649}a^{13}-\frac{2693983105368}{167422516077649}a^{12}-\frac{1535006455084}{23917502296807}a^{11}+\frac{11456814212692}{167422516077649}a^{10}+\frac{30145874047042}{167422516077649}a^{9}+\frac{35623775923739}{167422516077649}a^{8}-\frac{65431176039237}{167422516077649}a^{7}-\frac{25637024086860}{167422516077649}a^{6}+\frac{5428803808800}{23917502296807}a^{5}+\frac{71209597089950}{167422516077649}a^{4}-\frac{31630957886519}{167422516077649}a^{3}-\frac{9816613958478}{167422516077649}a^{2}-\frac{2815216711063}{23917502296807}a+\frac{18568929167}{3416786042401}$ 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:  $C_{3}$, which has order $3$
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:  $C_{3}$, which has order $3$
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:  $9$
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:   \( -\frac{3275753822}{16770761903} a^{19} + \frac{31119661309}{16770761903} a^{18} - \frac{112395367211}{16770761903} a^{17} + \frac{161809257914}{16770761903} a^{16} + \frac{127340035544}{16770761903} a^{15} - \frac{145289995906}{2395823129} a^{14} + \frac{1389804797950}{16770761903} a^{13} + \frac{2452761393550}{16770761903} a^{12} - \frac{10149628771638}{16770761903} a^{11} + \frac{106148191171}{188435527} a^{10} + \frac{4546568381501}{16770761903} a^{9} - \frac{11870774973098}{16770761903} a^{8} - \frac{625086058}{14880889} a^{7} + \frac{16288255564573}{16770761903} a^{6} - \frac{12486706382141}{16770761903} a^{5} - \frac{3400016538141}{16770761903} a^{4} + \frac{3874505029267}{16770761903} a^{3} + \frac{463181621528}{2395823129} a^{2} - \frac{54551052363}{342260447} a + \frac{8902526387}{342260447} \)  (order $6$) 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:   $\frac{6846783230104}{167422516077649}a^{19}-\frac{67221885902757}{167422516077649}a^{18}+\frac{36386140867277}{23917502296807}a^{17}-\frac{404671774620837}{167422516077649}a^{16}-\frac{186333624372941}{167422516077649}a^{15}+\frac{22\cdots 33}{167422516077649}a^{14}-\frac{35\cdots 81}{167422516077649}a^{13}-\frac{91149453747278}{3416786042401}a^{12}+\frac{23\cdots 85}{167422516077649}a^{11}-\frac{25\cdots 45}{167422516077649}a^{10}-\frac{57\cdots 03}{167422516077649}a^{9}+\frac{29\cdots 12}{167422516077649}a^{8}-\frac{44\cdots 01}{167422516077649}a^{7}-\frac{37\cdots 17}{167422516077649}a^{6}+\frac{35\cdots 28}{167422516077649}a^{5}+\frac{27\cdots 29}{167422516077649}a^{4}-\frac{12\cdots 06}{167422516077649}a^{3}-\frac{59\cdots 45}{167422516077649}a^{2}+\frac{11\cdots 01}{23917502296807}a-\frac{31659331077091}{3416786042401}$, $\frac{8326698335280}{167422516077649}a^{19}-\frac{80123737296199}{167422516077649}a^{18}+\frac{294625216710989}{167422516077649}a^{17}-\frac{439559075762454}{167422516077649}a^{16}-\frac{294750966098269}{167422516077649}a^{15}+\frac{26\cdots 71}{167422516077649}a^{14}-\frac{38\cdots 07}{167422516077649}a^{13}-\frac{60\cdots 38}{167422516077649}a^{12}+\frac{26\cdots 39}{167422516077649}a^{11}-\frac{26\cdots 61}{167422516077649}a^{10}-\frac{10\cdots 74}{167422516077649}a^{9}+\frac{32\cdots 89}{167422516077649}a^{8}-\frac{27576354104952}{23917502296807}a^{7}-\frac{43\cdots 59}{167422516077649}a^{6}+\frac{35\cdots 08}{167422516077649}a^{5}+\frac{75\cdots 53}{167422516077649}a^{4}-\frac{11\cdots 43}{167422516077649}a^{3}-\frac{85\cdots 23}{167422516077649}a^{2}+\frac{10\cdots 42}{23917502296807}a-\frac{26334843656490}{3416786042401}$, $\frac{16768309755655}{167422516077649}a^{19}-\frac{22603117428139}{23917502296807}a^{18}+\frac{565845170136203}{167422516077649}a^{17}-\frac{797857367256231}{167422516077649}a^{16}-\frac{683666168093512}{167422516077649}a^{15}+\frac{84249985411658}{2744631411109}a^{14}-\frac{68\cdots 32}{167422516077649}a^{13}-\frac{12\cdots 56}{167422516077649}a^{12}+\frac{50\cdots 09}{167422516077649}a^{11}-\frac{45\cdots 20}{167422516077649}a^{10}-\frac{24\cdots 40}{167422516077649}a^{9}+\frac{58\cdots 56}{167422516077649}a^{8}+\frac{62\cdots 38}{167422516077649}a^{7}-\frac{35\cdots 21}{7279239829463}a^{6}+\frac{59\cdots 57}{167422516077649}a^{5}+\frac{18\cdots 82}{167422516077649}a^{4}-\frac{17\cdots 61}{167422516077649}a^{3}-\frac{16\cdots 72}{167422516077649}a^{2}+\frac{18\cdots 60}{23917502296807}a-\frac{42923038973204}{3416786042401}$, $\frac{8326698335280}{167422516077649}a^{19}-\frac{1280057886461}{2744631411109}a^{18}+\frac{276263360712287}{167422516077649}a^{17}-\frac{378708488892232}{167422516077649}a^{16}-\frac{365353591756133}{167422516077649}a^{15}+\frac{25\cdots 24}{167422516077649}a^{14}-\frac{32\cdots 50}{167422516077649}a^{13}-\frac{942182356032599}{23917502296807}a^{12}+\frac{24\cdots 72}{167422516077649}a^{11}-\frac{30\cdots 80}{23917502296807}a^{10}-\frac{13\cdots 17}{167422516077649}a^{9}+\frac{27\cdots 80}{167422516077649}a^{8}+\frac{54\cdots 59}{167422516077649}a^{7}-\frac{58\cdots 11}{23917502296807}a^{6}+\frac{11\cdots 18}{7279239829463}a^{5}+\frac{11\cdots 15}{167422516077649}a^{4}-\frac{79\cdots 85}{167422516077649}a^{3}-\frac{95\cdots 64}{167422516077649}a^{2}+\frac{833205124462719}{23917502296807}a-\frac{9851858433478}{3416786042401}$, $\frac{4271163793385}{167422516077649}a^{19}-\frac{37792745444479}{167422516077649}a^{18}+\frac{121644619873158}{167422516077649}a^{17}-\frac{129035499599768}{167422516077649}a^{16}-\frac{260052464253203}{167422516077649}a^{15}+\frac{11\cdots 30}{167422516077649}a^{14}-\frac{10\cdots 38}{167422516077649}a^{13}-\frac{39\cdots 38}{167422516077649}a^{12}+\frac{10\cdots 56}{167422516077649}a^{11}-\frac{51\cdots 11}{167422516077649}a^{10}-\frac{10\cdots 36}{167422516077649}a^{9}+\frac{97\cdots 34}{167422516077649}a^{8}+\frac{326958414150824}{7279239829463}a^{7}-\frac{17\cdots 58}{167422516077649}a^{6}+\frac{49\cdots 59}{167422516077649}a^{5}+\frac{89\cdots 31}{167422516077649}a^{4}-\frac{238655919007452}{167422516077649}a^{3}-\frac{46\cdots 19}{167422516077649}a^{2}+\frac{8405743150079}{3416786042401}a+\frac{4022181991946}{3416786042401}$, $\frac{7246870194798}{167422516077649}a^{19}-\frac{68845266850581}{167422516077649}a^{18}+\frac{248678248858759}{167422516077649}a^{17}-\frac{358210810609636}{167422516077649}a^{16}-\frac{281351157524748}{167422516077649}a^{15}+\frac{22\cdots 68}{167422516077649}a^{14}-\frac{30\cdots 64}{167422516077649}a^{13}-\frac{54\cdots 14}{167422516077649}a^{12}+\frac{22\cdots 04}{167422516077649}a^{11}-\frac{235948784552085}{1881151866041}a^{10}-\frac{99\cdots 79}{167422516077649}a^{9}+\frac{26\cdots 71}{167422516077649}a^{8}+\frac{41732138121890}{7279239829463}a^{7}-\frac{35\cdots 46}{167422516077649}a^{6}+\frac{28\cdots 50}{167422516077649}a^{5}+\frac{69\cdots 79}{167422516077649}a^{4}-\frac{86\cdots 58}{167422516077649}a^{3}-\frac{66\cdots 10}{167422516077649}a^{2}+\frac{811401607565620}{23917502296807}a-\frac{18738403062281}{3416786042401}$, $\frac{63587703137606}{167422516077649}a^{19}-\frac{590015308065391}{167422516077649}a^{18}+\frac{33689430036557}{2744631411109}a^{17}-\frac{27\cdots 31}{167422516077649}a^{16}-\frac{29\cdots 11}{167422516077649}a^{15}+\frac{18\cdots 80}{167422516077649}a^{14}-\frac{23\cdots 74}{167422516077649}a^{13}-\frac{51\cdots 98}{167422516077649}a^{12}+\frac{18\cdots 62}{167422516077649}a^{11}-\frac{14\cdots 30}{167422516077649}a^{10}-\frac{11\cdots 09}{167422516077649}a^{9}+\frac{20\cdots 22}{167422516077649}a^{8}+\frac{49\cdots 88}{167422516077649}a^{7}-\frac{29\cdots 39}{167422516077649}a^{6}+\frac{18\cdots 61}{167422516077649}a^{5}+\frac{90\cdots 91}{167422516077649}a^{4}-\frac{49\cdots 07}{167422516077649}a^{3}-\frac{28\cdots 99}{7279239829463}a^{2}+\frac{51\cdots 80}{23917502296807}a-\frac{94674651612887}{3416786042401}$, $\frac{24579422544554}{167422516077649}a^{19}-\frac{239183830644421}{167422516077649}a^{18}+\frac{894404797129544}{167422516077649}a^{17}-\frac{197607467778925}{23917502296807}a^{16}-\frac{758642279350335}{167422516077649}a^{15}+\frac{79\cdots 04}{167422516077649}a^{14}-\frac{135132279037719}{1881151866041}a^{13}-\frac{16\cdots 17}{167422516077649}a^{12}+\frac{81\cdots 02}{167422516077649}a^{11}-\frac{85\cdots 04}{167422516077649}a^{10}-\frac{25\cdots 41}{167422516077649}a^{9}+\frac{10\cdots 57}{167422516077649}a^{8}-\frac{92\cdots 75}{167422516077649}a^{7}-\frac{18\cdots 03}{23917502296807}a^{6}+\frac{11\cdots 63}{167422516077649}a^{5}+\frac{22\cdots 35}{23917502296807}a^{4}-\frac{39\cdots 35}{167422516077649}a^{3}-\frac{22\cdots 59}{167422516077649}a^{2}+\frac{536008734317551}{3416786042401}a-\frac{98029380641088}{3416786042401}$, $\frac{960544963024}{23917502296807}a^{19}-\frac{64296416118463}{167422516077649}a^{18}+\frac{234631543684781}{167422516077649}a^{17}-\frac{346352445803579}{167422516077649}a^{16}-\frac{239321229097642}{167422516077649}a^{15}+\frac{299450134053520}{23917502296807}a^{14}-\frac{424148234918139}{23917502296807}a^{13}-\frac{54550418483447}{1881151866041}a^{12}+\frac{21\cdots 06}{167422516077649}a^{11}-\frac{20\cdots 67}{167422516077649}a^{10}-\frac{119135521996718}{2498843523547}a^{9}+\frac{24\cdots 60}{167422516077649}a^{8}+\frac{523777195942022}{167422516077649}a^{7}-\frac{33\cdots 86}{167422516077649}a^{6}+\frac{26\cdots 34}{167422516077649}a^{5}+\frac{52\cdots 22}{167422516077649}a^{4}-\frac{71\cdots 88}{167422516077649}a^{3}-\frac{984021302125210}{23917502296807}a^{2}+\frac{771539005406650}{23917502296807}a-\frac{16223282251398}{3416786042401}$ Copy content Toggle raw display
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:  \( 955501.823564 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 955501.823564 \cdot 3}{6\cdot\sqrt{35908028125401873392383429449}}\cr\approx \mathstrut & 0.241771124152 \end{aligned}\]

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^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49) 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^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49, 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^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49); 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^20 - 10*x^19 + 39*x^18 - 66*x^17 - 16*x^16 + 332*x^15 - 576*x^14 - 554*x^13 + 3490*x^12 - 4378*x^11 - 109*x^10 + 4411*x^9 - 1459*x^8 - 5238*x^7 + 6204*x^6 - 605*x^5 - 1802*x^4 - 524*x^3 + 1329*x^2 - 469*x + 49); 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

$D_{10}$ (as 20T4):

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)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-239}) \), \(\Q(\sqrt{717}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-239})\), 5.1.57121.1 x5, 10.0.779811265199.1, 10.2.189494137443357.1 x5, 10.0.792862499763.1 x5

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 10 siblings: 10.0.792862499763.1, 10.2.189494137443357.1
Minimal sibling: 10.0.792862499763.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ R ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.2.0.1}{2} }^{10}$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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
\(3\) Copy content Toggle raw display 3.5.2.5a1.2$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
3.5.2.5a1.2$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(239\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$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)