Properties

Label 21.9.778...553.1
Degree $21$
Signature $[9, 6]$
Discriminant $7.781\times 10^{28}$
Root discriminant \(23.76\)
Ramified primes $71,157,3709,8623$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^7.S_7$ (as 21T139)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1)
 
gp: K = bnfinit(y^21 - 8*y^20 + 15*y^19 + 42*y^18 - 181*y^17 + 43*y^16 + 664*y^15 - 785*y^14 - 999*y^13 + 2386*y^12 - 12*y^11 - 3337*y^10 + 2186*y^9 + 1818*y^8 - 2845*y^7 + 445*y^6 + 1349*y^5 - 754*y^4 - 347*y^3 + 279*y^2 + 76*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1)
 

\( x^{21} - 8 x^{20} + 15 x^{19} + 42 x^{18} - 181 x^{17} + 43 x^{16} + 664 x^{15} - 785 x^{14} - 999 x^{13} + 2386 x^{12} - 12 x^{11} - 3337 x^{10} + 2186 x^{9} + 1818 x^{8} - 2845 x^{7} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[9, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(77814973866654515283350056553\) \(\medspace = 71^{3}\cdot 157^{2}\cdot 3709^{2}\cdot 8623^{3}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(23.76\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $71^{1/2}157^{2/3}3709^{2/3}8623^{1/2}\approx 5456272.155977658$
Ramified primes:   \(71\), \(157\), \(3709\), \(8623\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{612233}) \)
$\card{ \Aut(K/\Q) }$:  $3$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{17\!\cdots\!99}a^{20}-\frac{21\!\cdots\!31}{17\!\cdots\!99}a^{19}-\frac{49\!\cdots\!65}{17\!\cdots\!99}a^{18}+\frac{82\!\cdots\!60}{17\!\cdots\!99}a^{17}+\frac{61\!\cdots\!17}{17\!\cdots\!99}a^{16}-\frac{45\!\cdots\!96}{17\!\cdots\!99}a^{15}-\frac{46\!\cdots\!59}{17\!\cdots\!99}a^{14}-\frac{86\!\cdots\!31}{17\!\cdots\!99}a^{13}-\frac{62\!\cdots\!83}{17\!\cdots\!99}a^{12}+\frac{23\!\cdots\!62}{17\!\cdots\!99}a^{11}-\frac{16\!\cdots\!94}{17\!\cdots\!99}a^{10}-\frac{76\!\cdots\!16}{17\!\cdots\!99}a^{9}-\frac{53\!\cdots\!34}{17\!\cdots\!99}a^{8}+\frac{77\!\cdots\!40}{17\!\cdots\!99}a^{7}-\frac{83\!\cdots\!94}{17\!\cdots\!99}a^{6}-\frac{84\!\cdots\!19}{17\!\cdots\!99}a^{5}+\frac{24\!\cdots\!62}{17\!\cdots\!99}a^{4}+\frac{26\!\cdots\!80}{17\!\cdots\!99}a^{3}+\frac{29\!\cdots\!72}{17\!\cdots\!99}a^{2}+\frac{34\!\cdots\!75}{17\!\cdots\!99}a-\frac{40\!\cdots\!34}{17\!\cdots\!99}$ Copy content Toggle raw display

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

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{81\!\cdots\!70}{17\!\cdots\!99}a^{20}-\frac{45\!\cdots\!54}{17\!\cdots\!99}a^{19}+\frac{45\!\cdots\!72}{17\!\cdots\!99}a^{18}+\frac{36\!\cdots\!52}{17\!\cdots\!99}a^{17}-\frac{37\!\cdots\!74}{17\!\cdots\!99}a^{16}-\frac{12\!\cdots\!90}{17\!\cdots\!99}a^{15}+\frac{16\!\cdots\!04}{17\!\cdots\!99}a^{14}+\frac{27\!\cdots\!26}{17\!\cdots\!99}a^{13}-\frac{39\!\cdots\!03}{17\!\cdots\!99}a^{12}-\frac{50\!\cdots\!52}{17\!\cdots\!99}a^{11}+\frac{74\!\cdots\!87}{17\!\cdots\!99}a^{10}+\frac{66\!\cdots\!73}{17\!\cdots\!99}a^{9}-\frac{12\!\cdots\!26}{17\!\cdots\!99}a^{8}-\frac{31\!\cdots\!78}{17\!\cdots\!99}a^{7}+\frac{15\!\cdots\!98}{17\!\cdots\!99}a^{6}-\frac{73\!\cdots\!63}{17\!\cdots\!99}a^{5}-\frac{78\!\cdots\!37}{17\!\cdots\!99}a^{4}+\frac{11\!\cdots\!75}{17\!\cdots\!99}a^{3}-\frac{28\!\cdots\!54}{17\!\cdots\!99}a^{2}-\frac{36\!\cdots\!19}{17\!\cdots\!99}a+\frac{99\!\cdots\!00}{17\!\cdots\!99}$, $\frac{92\!\cdots\!89}{17\!\cdots\!99}a^{20}-\frac{67\!\cdots\!08}{17\!\cdots\!99}a^{19}+\frac{10\!\cdots\!93}{17\!\cdots\!99}a^{18}+\frac{41\!\cdots\!59}{17\!\cdots\!99}a^{17}-\frac{13\!\cdots\!89}{17\!\cdots\!99}a^{16}-\frac{20\!\cdots\!28}{17\!\cdots\!99}a^{15}+\frac{54\!\cdots\!76}{17\!\cdots\!99}a^{14}-\frac{44\!\cdots\!00}{17\!\cdots\!99}a^{13}-\frac{94\!\cdots\!05}{17\!\cdots\!99}a^{12}+\frac{15\!\cdots\!06}{17\!\cdots\!99}a^{11}+\frac{42\!\cdots\!59}{17\!\cdots\!99}a^{10}-\frac{24\!\cdots\!69}{17\!\cdots\!99}a^{9}+\frac{11\!\cdots\!39}{17\!\cdots\!99}a^{8}+\frac{15\!\cdots\!26}{17\!\cdots\!99}a^{7}-\frac{18\!\cdots\!39}{17\!\cdots\!99}a^{6}+\frac{70\!\cdots\!39}{17\!\cdots\!99}a^{5}+\frac{10\!\cdots\!63}{17\!\cdots\!99}a^{4}-\frac{46\!\cdots\!84}{17\!\cdots\!99}a^{3}-\frac{23\!\cdots\!38}{17\!\cdots\!99}a^{2}+\frac{15\!\cdots\!77}{17\!\cdots\!99}a+\frac{19\!\cdots\!41}{17\!\cdots\!99}$, $\frac{81\!\cdots\!39}{17\!\cdots\!99}a^{20}-\frac{65\!\cdots\!10}{17\!\cdots\!99}a^{19}+\frac{12\!\cdots\!05}{17\!\cdots\!99}a^{18}+\frac{31\!\cdots\!41}{17\!\cdots\!99}a^{17}-\frac{14\!\cdots\!68}{17\!\cdots\!99}a^{16}+\frac{56\!\cdots\!20}{17\!\cdots\!99}a^{15}+\frac{49\!\cdots\!66}{17\!\cdots\!99}a^{14}-\frac{68\!\cdots\!40}{17\!\cdots\!99}a^{13}-\frac{62\!\cdots\!47}{17\!\cdots\!99}a^{12}+\frac{19\!\cdots\!40}{17\!\cdots\!99}a^{11}-\frac{38\!\cdots\!57}{17\!\cdots\!99}a^{10}-\frac{24\!\cdots\!21}{17\!\cdots\!99}a^{9}+\frac{21\!\cdots\!53}{17\!\cdots\!99}a^{8}+\frac{95\!\cdots\!86}{17\!\cdots\!99}a^{7}-\frac{22\!\cdots\!15}{17\!\cdots\!99}a^{6}+\frac{76\!\cdots\!31}{17\!\cdots\!99}a^{5}+\frac{85\!\cdots\!72}{17\!\cdots\!99}a^{4}-\frac{70\!\cdots\!84}{17\!\cdots\!99}a^{3}-\frac{13\!\cdots\!38}{17\!\cdots\!99}a^{2}+\frac{23\!\cdots\!32}{17\!\cdots\!99}a+\frac{16\!\cdots\!05}{17\!\cdots\!99}$, $\frac{71\!\cdots\!84}{17\!\cdots\!99}a^{20}-\frac{55\!\cdots\!80}{17\!\cdots\!99}a^{19}+\frac{10\!\cdots\!76}{17\!\cdots\!99}a^{18}+\frac{27\!\cdots\!04}{17\!\cdots\!99}a^{17}-\frac{11\!\cdots\!13}{17\!\cdots\!99}a^{16}+\frac{38\!\cdots\!60}{17\!\cdots\!99}a^{15}+\frac{39\!\cdots\!34}{17\!\cdots\!99}a^{14}-\frac{52\!\cdots\!04}{17\!\cdots\!99}a^{13}-\frac{47\!\cdots\!31}{17\!\cdots\!99}a^{12}+\frac{14\!\cdots\!90}{17\!\cdots\!99}a^{11}-\frac{35\!\cdots\!45}{17\!\cdots\!99}a^{10}-\frac{16\!\cdots\!31}{17\!\cdots\!99}a^{9}+\frac{16\!\cdots\!22}{17\!\cdots\!99}a^{8}+\frac{44\!\cdots\!00}{17\!\cdots\!99}a^{7}-\frac{15\!\cdots\!36}{17\!\cdots\!99}a^{6}+\frac{73\!\cdots\!83}{17\!\cdots\!99}a^{5}+\frac{42\!\cdots\!67}{17\!\cdots\!99}a^{4}-\frac{46\!\cdots\!73}{17\!\cdots\!99}a^{3}-\frac{37\!\cdots\!51}{17\!\cdots\!99}a^{2}+\frac{12\!\cdots\!79}{17\!\cdots\!99}a-\frac{10\!\cdots\!75}{17\!\cdots\!99}$, $\frac{81\!\cdots\!70}{17\!\cdots\!99}a^{20}-\frac{45\!\cdots\!54}{17\!\cdots\!99}a^{19}+\frac{45\!\cdots\!72}{17\!\cdots\!99}a^{18}+\frac{36\!\cdots\!52}{17\!\cdots\!99}a^{17}-\frac{37\!\cdots\!74}{17\!\cdots\!99}a^{16}-\frac{12\!\cdots\!90}{17\!\cdots\!99}a^{15}+\frac{16\!\cdots\!04}{17\!\cdots\!99}a^{14}+\frac{27\!\cdots\!26}{17\!\cdots\!99}a^{13}-\frac{39\!\cdots\!03}{17\!\cdots\!99}a^{12}-\frac{50\!\cdots\!52}{17\!\cdots\!99}a^{11}+\frac{74\!\cdots\!87}{17\!\cdots\!99}a^{10}+\frac{66\!\cdots\!73}{17\!\cdots\!99}a^{9}-\frac{12\!\cdots\!26}{17\!\cdots\!99}a^{8}-\frac{31\!\cdots\!78}{17\!\cdots\!99}a^{7}+\frac{15\!\cdots\!98}{17\!\cdots\!99}a^{6}-\frac{73\!\cdots\!63}{17\!\cdots\!99}a^{5}-\frac{78\!\cdots\!37}{17\!\cdots\!99}a^{4}+\frac{11\!\cdots\!75}{17\!\cdots\!99}a^{3}-\frac{28\!\cdots\!54}{17\!\cdots\!99}a^{2}-\frac{18\!\cdots\!20}{17\!\cdots\!99}a+\frac{99\!\cdots\!00}{17\!\cdots\!99}$, $\frac{61\!\cdots\!75}{17\!\cdots\!99}a^{20}-\frac{43\!\cdots\!69}{17\!\cdots\!99}a^{19}+\frac{57\!\cdots\!04}{17\!\cdots\!99}a^{18}+\frac{26\!\cdots\!82}{17\!\cdots\!99}a^{17}-\frac{80\!\cdots\!13}{17\!\cdots\!99}a^{16}-\frac{19\!\cdots\!49}{17\!\cdots\!99}a^{15}+\frac{30\!\cdots\!45}{17\!\cdots\!99}a^{14}-\frac{22\!\cdots\!77}{17\!\cdots\!99}a^{13}-\frac{48\!\cdots\!31}{17\!\cdots\!99}a^{12}+\frac{77\!\cdots\!75}{17\!\cdots\!99}a^{11}+\frac{13\!\cdots\!81}{17\!\cdots\!99}a^{10}-\frac{10\!\cdots\!04}{17\!\cdots\!99}a^{9}+\frac{63\!\cdots\!79}{17\!\cdots\!99}a^{8}+\frac{44\!\cdots\!11}{17\!\cdots\!99}a^{7}-\frac{77\!\cdots\!11}{17\!\cdots\!99}a^{6}+\frac{24\!\cdots\!62}{17\!\cdots\!99}a^{5}+\frac{21\!\cdots\!54}{17\!\cdots\!99}a^{4}-\frac{15\!\cdots\!47}{17\!\cdots\!99}a^{3}-\frac{20\!\cdots\!94}{17\!\cdots\!99}a^{2}-\frac{66\!\cdots\!77}{17\!\cdots\!99}a-\frac{13\!\cdots\!47}{17\!\cdots\!99}$, $\frac{86\!\cdots\!15}{17\!\cdots\!99}a^{20}+\frac{31\!\cdots\!95}{17\!\cdots\!99}a^{19}-\frac{54\!\cdots\!77}{17\!\cdots\!99}a^{18}+\frac{10\!\cdots\!96}{17\!\cdots\!99}a^{17}+\frac{32\!\cdots\!09}{17\!\cdots\!99}a^{16}-\frac{11\!\cdots\!67}{17\!\cdots\!99}a^{15}-\frac{34\!\cdots\!48}{17\!\cdots\!99}a^{14}+\frac{42\!\cdots\!27}{17\!\cdots\!99}a^{13}-\frac{22\!\cdots\!41}{17\!\cdots\!99}a^{12}-\frac{81\!\cdots\!68}{17\!\cdots\!99}a^{11}+\frac{93\!\cdots\!55}{17\!\cdots\!99}a^{10}+\frac{67\!\cdots\!03}{17\!\cdots\!99}a^{9}-\frac{15\!\cdots\!94}{17\!\cdots\!99}a^{8}+\frac{21\!\cdots\!19}{17\!\cdots\!99}a^{7}+\frac{12\!\cdots\!53}{17\!\cdots\!99}a^{6}-\frac{84\!\cdots\!78}{17\!\cdots\!99}a^{5}-\frac{29\!\cdots\!69}{17\!\cdots\!99}a^{4}+\frac{60\!\cdots\!68}{17\!\cdots\!99}a^{3}-\frac{64\!\cdots\!85}{17\!\cdots\!99}a^{2}-\frac{21\!\cdots\!17}{17\!\cdots\!99}a+\frac{99\!\cdots\!44}{17\!\cdots\!99}$, $\frac{17\!\cdots\!76}{17\!\cdots\!99}a^{20}-\frac{14\!\cdots\!28}{17\!\cdots\!99}a^{19}+\frac{26\!\cdots\!36}{17\!\cdots\!99}a^{18}+\frac{69\!\cdots\!38}{17\!\cdots\!99}a^{17}-\frac{31\!\cdots\!29}{17\!\cdots\!99}a^{16}+\frac{10\!\cdots\!50}{17\!\cdots\!99}a^{15}+\frac{10\!\cdots\!42}{17\!\cdots\!99}a^{14}-\frac{14\!\cdots\!67}{17\!\cdots\!99}a^{13}-\frac{14\!\cdots\!74}{17\!\cdots\!99}a^{12}+\frac{40\!\cdots\!46}{17\!\cdots\!99}a^{11}-\frac{73\!\cdots\!26}{17\!\cdots\!99}a^{10}-\frac{51\!\cdots\!13}{17\!\cdots\!99}a^{9}+\frac{44\!\cdots\!09}{17\!\cdots\!99}a^{8}+\frac{19\!\cdots\!25}{17\!\cdots\!99}a^{7}-\frac{48\!\cdots\!82}{17\!\cdots\!99}a^{6}+\frac{17\!\cdots\!78}{17\!\cdots\!99}a^{5}+\frac{16\!\cdots\!82}{17\!\cdots\!99}a^{4}-\frac{14\!\cdots\!69}{17\!\cdots\!99}a^{3}-\frac{19\!\cdots\!73}{17\!\cdots\!99}a^{2}+\frac{43\!\cdots\!20}{17\!\cdots\!99}a+\frac{15\!\cdots\!47}{17\!\cdots\!99}$, $\frac{73\!\cdots\!82}{17\!\cdots\!99}a^{20}-\frac{61\!\cdots\!76}{17\!\cdots\!99}a^{19}+\frac{12\!\cdots\!06}{17\!\cdots\!99}a^{18}+\frac{27\!\cdots\!84}{17\!\cdots\!99}a^{17}-\frac{14\!\cdots\!26}{17\!\cdots\!99}a^{16}+\frac{72\!\cdots\!62}{17\!\cdots\!99}a^{15}+\frac{49\!\cdots\!88}{17\!\cdots\!99}a^{14}-\frac{73\!\cdots\!43}{17\!\cdots\!99}a^{13}-\frac{59\!\cdots\!39}{17\!\cdots\!99}a^{12}+\frac{20\!\cdots\!24}{17\!\cdots\!99}a^{11}-\frac{48\!\cdots\!89}{17\!\cdots\!99}a^{10}-\frac{25\!\cdots\!18}{17\!\cdots\!99}a^{9}+\frac{22\!\cdots\!86}{17\!\cdots\!99}a^{8}+\frac{93\!\cdots\!89}{17\!\cdots\!99}a^{7}-\frac{23\!\cdots\!54}{17\!\cdots\!99}a^{6}+\frac{81\!\cdots\!73}{17\!\cdots\!99}a^{5}+\frac{83\!\cdots\!00}{17\!\cdots\!99}a^{4}-\frac{68\!\cdots\!24}{17\!\cdots\!99}a^{3}-\frac{14\!\cdots\!65}{17\!\cdots\!99}a^{2}+\frac{22\!\cdots\!85}{17\!\cdots\!99}a+\frac{32\!\cdots\!28}{17\!\cdots\!99}$, $\frac{13\!\cdots\!92}{17\!\cdots\!99}a^{20}-\frac{11\!\cdots\!38}{17\!\cdots\!99}a^{19}+\frac{21\!\cdots\!75}{17\!\cdots\!99}a^{18}+\frac{54\!\cdots\!88}{17\!\cdots\!99}a^{17}-\frac{25\!\cdots\!69}{17\!\cdots\!99}a^{16}+\frac{85\!\cdots\!32}{17\!\cdots\!99}a^{15}+\frac{87\!\cdots\!91}{17\!\cdots\!99}a^{14}-\frac{11\!\cdots\!07}{17\!\cdots\!99}a^{13}-\frac{11\!\cdots\!87}{17\!\cdots\!99}a^{12}+\frac{32\!\cdots\!33}{17\!\cdots\!99}a^{11}-\frac{47\!\cdots\!47}{17\!\cdots\!99}a^{10}-\frac{42\!\cdots\!73}{17\!\cdots\!99}a^{9}+\frac{34\!\cdots\!45}{17\!\cdots\!99}a^{8}+\frac{18\!\cdots\!21}{17\!\cdots\!99}a^{7}-\frac{39\!\cdots\!42}{17\!\cdots\!99}a^{6}+\frac{12\!\cdots\!21}{17\!\cdots\!99}a^{5}+\frac{14\!\cdots\!14}{17\!\cdots\!99}a^{4}-\frac{12\!\cdots\!39}{17\!\cdots\!99}a^{3}-\frac{22\!\cdots\!73}{17\!\cdots\!99}a^{2}+\frac{38\!\cdots\!69}{17\!\cdots\!99}a+\frac{29\!\cdots\!72}{17\!\cdots\!99}$, $\frac{36\!\cdots\!31}{17\!\cdots\!99}a^{20}-\frac{20\!\cdots\!01}{17\!\cdots\!99}a^{19}-\frac{18\!\cdots\!25}{17\!\cdots\!99}a^{18}+\frac{20\!\cdots\!04}{17\!\cdots\!99}a^{17}-\frac{25\!\cdots\!82}{17\!\cdots\!99}a^{16}-\frac{75\!\cdots\!33}{17\!\cdots\!99}a^{15}+\frac{16\!\cdots\!43}{17\!\cdots\!99}a^{14}+\frac{10\!\cdots\!58}{17\!\cdots\!99}a^{13}-\frac{43\!\cdots\!35}{17\!\cdots\!99}a^{12}+\frac{72\!\cdots\!84}{17\!\cdots\!99}a^{11}+\frac{61\!\cdots\!17}{17\!\cdots\!99}a^{10}-\frac{42\!\cdots\!68}{17\!\cdots\!99}a^{9}-\frac{36\!\cdots\!00}{17\!\cdots\!99}a^{8}+\frac{52\!\cdots\!18}{17\!\cdots\!99}a^{7}-\frac{65\!\cdots\!67}{17\!\cdots\!99}a^{6}-\frac{22\!\cdots\!03}{17\!\cdots\!99}a^{5}+\frac{11\!\cdots\!66}{17\!\cdots\!99}a^{4}+\frac{57\!\cdots\!11}{17\!\cdots\!99}a^{3}-\frac{47\!\cdots\!65}{17\!\cdots\!99}a^{2}-\frac{79\!\cdots\!49}{17\!\cdots\!99}a-\frac{50\!\cdots\!69}{17\!\cdots\!99}$, $\frac{83\!\cdots\!95}{17\!\cdots\!99}a^{20}-\frac{66\!\cdots\!53}{17\!\cdots\!99}a^{19}+\frac{12\!\cdots\!36}{17\!\cdots\!99}a^{18}+\frac{34\!\cdots\!03}{17\!\cdots\!99}a^{17}-\frac{15\!\cdots\!71}{17\!\cdots\!99}a^{16}+\frac{40\!\cdots\!02}{17\!\cdots\!99}a^{15}+\frac{54\!\cdots\!84}{17\!\cdots\!99}a^{14}-\frac{67\!\cdots\!98}{17\!\cdots\!99}a^{13}-\frac{79\!\cdots\!97}{17\!\cdots\!99}a^{12}+\frac{20\!\cdots\!71}{17\!\cdots\!99}a^{11}-\frac{13\!\cdots\!22}{17\!\cdots\!99}a^{10}-\frac{28\!\cdots\!77}{17\!\cdots\!99}a^{9}+\frac{20\!\cdots\!19}{17\!\cdots\!99}a^{8}+\frac{14\!\cdots\!80}{17\!\cdots\!99}a^{7}-\frac{25\!\cdots\!96}{17\!\cdots\!99}a^{6}+\frac{50\!\cdots\!93}{17\!\cdots\!99}a^{5}+\frac{11\!\cdots\!43}{17\!\cdots\!99}a^{4}-\frac{68\!\cdots\!96}{17\!\cdots\!99}a^{3}-\frac{25\!\cdots\!95}{17\!\cdots\!99}a^{2}+\frac{23\!\cdots\!91}{17\!\cdots\!99}a+\frac{51\!\cdots\!86}{17\!\cdots\!99}$, $\frac{17\!\cdots\!29}{17\!\cdots\!99}a^{20}-\frac{70\!\cdots\!43}{17\!\cdots\!99}a^{19}-\frac{15\!\cdots\!63}{17\!\cdots\!99}a^{18}+\frac{89\!\cdots\!58}{17\!\cdots\!99}a^{17}+\frac{45\!\cdots\!92}{17\!\cdots\!99}a^{16}-\frac{53\!\cdots\!50}{17\!\cdots\!99}a^{15}+\frac{78\!\cdots\!93}{17\!\cdots\!99}a^{14}+\frac{17\!\cdots\!10}{17\!\cdots\!99}a^{13}-\frac{10\!\cdots\!85}{17\!\cdots\!99}a^{12}-\frac{36\!\cdots\!46}{17\!\cdots\!99}a^{11}+\frac{33\!\cdots\!14}{17\!\cdots\!99}a^{10}+\frac{41\!\cdots\!61}{17\!\cdots\!99}a^{9}-\frac{59\!\cdots\!26}{17\!\cdots\!99}a^{8}-\frac{16\!\cdots\!40}{17\!\cdots\!99}a^{7}+\frac{57\!\cdots\!58}{17\!\cdots\!99}a^{6}-\frac{14\!\cdots\!32}{17\!\cdots\!99}a^{5}-\frac{27\!\cdots\!55}{17\!\cdots\!99}a^{4}+\frac{19\!\cdots\!82}{17\!\cdots\!99}a^{3}+\frac{68\!\cdots\!00}{17\!\cdots\!99}a^{2}-\frac{88\!\cdots\!36}{17\!\cdots\!99}a-\frac{35\!\cdots\!84}{17\!\cdots\!99}$, $\frac{77\!\cdots\!32}{17\!\cdots\!99}a^{20}-\frac{57\!\cdots\!59}{17\!\cdots\!99}a^{19}+\frac{86\!\cdots\!21}{17\!\cdots\!99}a^{18}+\frac{33\!\cdots\!19}{17\!\cdots\!99}a^{17}-\frac{11\!\cdots\!96}{17\!\cdots\!99}a^{16}-\frac{49\!\cdots\!71}{17\!\cdots\!99}a^{15}+\frac{43\!\cdots\!87}{17\!\cdots\!99}a^{14}-\frac{40\!\cdots\!99}{17\!\cdots\!99}a^{13}-\frac{68\!\cdots\!44}{17\!\cdots\!99}a^{12}+\frac{13\!\cdots\!67}{17\!\cdots\!99}a^{11}+\frac{95\!\cdots\!46}{17\!\cdots\!99}a^{10}-\frac{18\!\cdots\!40}{17\!\cdots\!99}a^{9}+\frac{12\!\cdots\!27}{17\!\cdots\!99}a^{8}+\frac{83\!\cdots\!88}{17\!\cdots\!99}a^{7}-\frac{15\!\cdots\!88}{17\!\cdots\!99}a^{6}+\frac{49\!\cdots\!55}{17\!\cdots\!99}a^{5}+\frac{57\!\cdots\!52}{17\!\cdots\!99}a^{4}-\frac{49\!\cdots\!31}{17\!\cdots\!99}a^{3}-\frac{78\!\cdots\!48}{17\!\cdots\!99}a^{2}+\frac{14\!\cdots\!72}{17\!\cdots\!99}a-\frac{26\!\cdots\!67}{17\!\cdots\!99}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 3400600.48543 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot 3400600.48543 \cdot 1}{2\cdot\sqrt{77814973866654515283350056553}}\cr\approx \mathstrut & 0.192018556240 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 8*x^20 + 15*x^19 + 42*x^18 - 181*x^17 + 43*x^16 + 664*x^15 - 785*x^14 - 999*x^13 + 2386*x^12 - 12*x^11 - 3337*x^10 + 2186*x^9 + 1818*x^8 - 2845*x^7 + 445*x^6 + 1349*x^5 - 754*x^4 - 347*x^3 + 279*x^2 + 76*x + 1);
 
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

$C_3^7.S_7$ (as 21T139):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 11022480
The 429 conjugacy class representatives for $C_3^7.S_7$ are not computed
Character table for $C_3^7.S_7$ is not computed

Intermediate fields

7.3.612233.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 21 sibling: data not computed
Degree 42 siblings: 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 $21$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{3}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$ $18{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ $18{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.5.0.1}{5} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{3}$ $15{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(71\) Copy content Toggle raw display 71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.5.0.1$x^{5} + 18 x + 64$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} + 18 x + 64$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} + 18 x + 64$$1$$5$$0$$C_5$$[\ ]^{5}$
\(157\) Copy content Toggle raw display 157.3.2.2$x^{3} + 471$$3$$1$$2$$C_3$$[\ ]_{3}$
157.3.0.1$x^{3} + x + 152$$1$$3$$0$$C_3$$[\ ]^{3}$
157.15.0.1$x^{15} - x + 38$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(3709\) Copy content Toggle raw display $\Q_{3709}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{3709}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{3709}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$3$$1$$2$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(8623\) Copy content Toggle raw display Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$2$$3$$3$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$