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Properties

Label 21.21.941...237.1
Degree $21$
Signature $[21, 0]$
Discriminant $9.414\times 10^{31}$
Root discriminant \(33.31\)
Ramified primes $7,173$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7$ (as 21T4)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*x + 1)
 
Copy content gp:K = bnfinit(y^21 - 31*y^19 - 3*y^18 + 362*y^17 + 32*y^16 - 2119*y^15 - 158*y^14 + 6826*y^13 + 676*y^12 - 12509*y^11 - 2021*y^10 + 12809*y^9 + 3175*y^8 - 6710*y^7 - 2254*y^6 + 1442*y^5 + 571*y^4 - 97*y^3 - 44*y^2 + 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*x + 1)
 

\( x^{21} - 31 x^{19} - 3 x^{18} + 362 x^{17} + 32 x^{16} - 2119 x^{15} - 158 x^{14} + 6826 x^{13} + \cdots + 1 \) 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:  $21$
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:  $[21, 0]$
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:   \(94142881806955162927406195366237\) \(\medspace = 7^{14}\cdot 173^{9}\) 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:  \(33.31\)
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:  $7^{2/3}173^{1/2}\approx 48.13065200407494$
Ramified primes:   \(7\), \(173\) 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:  \(\Q(\sqrt{173}) \)
$\Aut(K/\Q)$:   $C_3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{18}+\frac{3}{14}a^{17}+\frac{3}{14}a^{16}-\frac{1}{7}a^{15}-\frac{1}{14}a^{13}-\frac{3}{14}a^{12}+\frac{2}{7}a^{11}+\frac{3}{14}a^{9}+\frac{3}{7}a^{8}-\frac{3}{14}a^{7}+\frac{3}{14}a^{6}+\frac{5}{14}a^{4}-\frac{3}{14}a^{3}+\frac{2}{7}a^{2}-\frac{5}{14}a+\frac{3}{7}$, $\frac{1}{14}a^{19}+\frac{1}{14}a^{17}+\frac{3}{14}a^{16}-\frac{1}{14}a^{15}-\frac{1}{14}a^{14}+\frac{3}{7}a^{12}+\frac{1}{7}a^{11}+\frac{3}{14}a^{10}+\frac{2}{7}a^{9}-\frac{1}{2}a^{8}+\frac{5}{14}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{3}{14}a^{4}-\frac{1}{14}a^{3}-\frac{3}{14}a^{2}+\frac{3}{14}$, $\frac{1}{162934304791286}a^{20}-\frac{88026754681}{162934304791286}a^{19}+\frac{770132756654}{81467152395643}a^{18}-\frac{2607635156019}{11638164627949}a^{17}+\frac{1751212016901}{23276329255898}a^{16}-\frac{16170348231445}{162934304791286}a^{15}-\frac{31896696514475}{162934304791286}a^{14}+\frac{23609362288305}{162934304791286}a^{13}+\frac{27042954172520}{81467152395643}a^{12}+\frac{78251361463763}{162934304791286}a^{11}+\frac{51359584672889}{162934304791286}a^{10}-\frac{34741355338729}{162934304791286}a^{9}-\frac{37116913859171}{81467152395643}a^{8}+\frac{25558031868333}{162934304791286}a^{7}+\frac{25550390568197}{81467152395643}a^{6}+\frac{6305989792553}{81467152395643}a^{5}+\frac{14719118160162}{81467152395643}a^{4}+\frac{1046377285263}{23276329255898}a^{3}-\frac{31921751560399}{162934304791286}a^{2}-\frac{50861426584869}{162934304791286}a+\frac{20693501531267}{81467152395643}$ 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:  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:  Trivial group, which has order $1$ (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:  $20$
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:   $\frac{241736426362034}{81467152395643}a^{20}-\frac{97729167094114}{81467152395643}a^{19}-\frac{10\cdots 07}{11638164627949}a^{18}+\frac{23\cdots 84}{81467152395643}a^{17}+\frac{86\cdots 20}{81467152395643}a^{16}-\frac{27\cdots 79}{81467152395643}a^{15}-\frac{50\cdots 03}{81467152395643}a^{14}+\frac{24\cdots 01}{11638164627949}a^{13}+\frac{16\cdots 23}{81467152395643}a^{12}-\frac{50\cdots 51}{81467152395643}a^{11}-\frac{28\cdots 95}{81467152395643}a^{10}+\frac{72\cdots 60}{81467152395643}a^{9}+\frac{29\cdots 77}{81467152395643}a^{8}-\frac{46\cdots 19}{81467152395643}a^{7}-\frac{15\cdots 43}{81467152395643}a^{6}+\frac{10\cdots 57}{81467152395643}a^{5}+\frac{53\cdots 98}{11638164627949}a^{4}-\frac{74\cdots 44}{81467152395643}a^{3}-\frac{33\cdots 23}{81467152395643}a^{2}+\frac{169777123424575}{81467152395643}a+\frac{129102919564606}{11638164627949}$, $\frac{533724032552540}{81467152395643}a^{20}-\frac{24105479043471}{11638164627949}a^{19}-\frac{16\cdots 01}{81467152395643}a^{18}+\frac{36\cdots 25}{81467152395643}a^{17}+\frac{19\cdots 66}{81467152395643}a^{16}-\frac{43\cdots 60}{81467152395643}a^{15}-\frac{15\cdots 09}{11638164627949}a^{14}+\frac{26\cdots 81}{81467152395643}a^{13}+\frac{35\cdots 21}{81467152395643}a^{12}-\frac{76\cdots 00}{81467152395643}a^{11}-\frac{63\cdots 52}{81467152395643}a^{10}+\frac{95\cdots 11}{81467152395643}a^{9}+\frac{63\cdots 52}{81467152395643}a^{8}-\frac{36\cdots 83}{81467152395643}a^{7}-\frac{33\cdots 85}{81467152395643}a^{6}-\frac{10\cdots 16}{81467152395643}a^{5}+\frac{73\cdots 37}{81467152395643}a^{4}+\frac{48\cdots 52}{81467152395643}a^{3}-\frac{54\cdots 31}{81467152395643}a^{2}-\frac{260155958644319}{11638164627949}a+\frac{11\cdots 94}{81467152395643}$, $\frac{148352081158277}{81467152395643}a^{20}-\frac{204895834248407}{162934304791286}a^{19}-\frac{45\cdots 85}{81467152395643}a^{18}+\frac{54\cdots 11}{162934304791286}a^{17}+\frac{10\cdots 81}{162934304791286}a^{16}-\frac{64\cdots 61}{162934304791286}a^{15}-\frac{61\cdots 07}{162934304791286}a^{14}+\frac{18\cdots 97}{81467152395643}a^{13}+\frac{95\cdots 22}{81467152395643}a^{12}-\frac{57\cdots 96}{81467152395643}a^{11}-\frac{48\cdots 23}{23276329255898}a^{10}+\frac{90\cdots 49}{81467152395643}a^{9}+\frac{34\cdots 87}{162934304791286}a^{8}-\frac{14\cdots 77}{162934304791286}a^{7}-\frac{93\cdots 54}{81467152395643}a^{6}+\frac{28\cdots 59}{81467152395643}a^{5}+\frac{48\cdots 17}{162934304791286}a^{4}-\frac{94\cdots 11}{162934304791286}a^{3}-\frac{52\cdots 95}{162934304791286}a^{2}+\frac{22\cdots 55}{81467152395643}a+\frac{203736323938155}{23276329255898}$, $a$, $\frac{635276877148379}{162934304791286}a^{20}-\frac{151543519019391}{162934304791286}a^{19}-\frac{19\cdots 93}{162934304791286}a^{18}+\frac{399438600027681}{23276329255898}a^{17}+\frac{11\cdots 03}{81467152395643}a^{16}-\frac{49\cdots 19}{23276329255898}a^{15}-\frac{13\cdots 27}{162934304791286}a^{14}+\frac{11\cdots 49}{81467152395643}a^{13}+\frac{42\cdots 81}{162934304791286}a^{12}-\frac{61\cdots 35}{162934304791286}a^{11}-\frac{76\cdots 69}{162934304791286}a^{10}+\frac{31\cdots 25}{81467152395643}a^{9}+\frac{55\cdots 39}{11638164627949}a^{8}+\frac{18\cdots 33}{81467152395643}a^{7}-\frac{57\cdots 61}{23276329255898}a^{6}-\frac{17\cdots 29}{81467152395643}a^{5}+\frac{88\cdots 07}{162934304791286}a^{4}+\frac{72\cdots 46}{11638164627949}a^{3}-\frac{63\cdots 05}{162934304791286}a^{2}-\frac{19\cdots 26}{81467152395643}a+\frac{635971238936020}{81467152395643}$, $\frac{402726592036711}{81467152395643}a^{20}-\frac{262311109222115}{162934304791286}a^{19}-\frac{24\cdots 59}{162934304791286}a^{18}+\frac{405005621405930}{11638164627949}a^{17}+\frac{14\cdots 70}{81467152395643}a^{16}-\frac{68\cdots 05}{162934304791286}a^{15}-\frac{16\cdots 17}{162934304791286}a^{14}+\frac{41\cdots 89}{162934304791286}a^{13}+\frac{52\cdots 27}{162934304791286}a^{12}-\frac{58\cdots 27}{81467152395643}a^{11}-\frac{94\cdots 55}{162934304791286}a^{10}+\frac{14\cdots 75}{162934304791286}a^{9}+\frac{94\cdots 75}{162934304791286}a^{8}-\frac{25\cdots 61}{81467152395643}a^{7}-\frac{49\cdots 89}{162934304791286}a^{6}-\frac{10\cdots 86}{81467152395643}a^{5}+\frac{54\cdots 29}{81467152395643}a^{4}+\frac{66\cdots 75}{11638164627949}a^{3}-\frac{84\cdots 81}{162934304791286}a^{2}-\frac{44\cdots 89}{162934304791286}a+\frac{20\cdots 29}{162934304791286}$, $\frac{309853210687116}{81467152395643}a^{20}-\frac{139365314062983}{81467152395643}a^{19}-\frac{95\cdots 34}{81467152395643}a^{18}+\frac{33\cdots 30}{81467152395643}a^{17}+\frac{22\cdots 51}{162934304791286}a^{16}-\frac{39\cdots 25}{81467152395643}a^{15}-\frac{64\cdots 62}{81467152395643}a^{14}+\frac{23\cdots 26}{81467152395643}a^{13}+\frac{29\cdots 34}{11638164627949}a^{12}-\frac{13\cdots 65}{162934304791286}a^{11}-\frac{36\cdots 60}{81467152395643}a^{10}+\frac{28\cdots 91}{23276329255898}a^{9}+\frac{73\cdots 69}{162934304791286}a^{8}-\frac{12\cdots 69}{162934304791286}a^{7}-\frac{19\cdots 45}{81467152395643}a^{6}+\frac{13\cdots 68}{81467152395643}a^{5}+\frac{86\cdots 15}{162934304791286}a^{4}-\frac{18\cdots 95}{162934304791286}a^{3}-\frac{68\cdots 19}{162934304791286}a^{2}+\frac{14\cdots 45}{162934304791286}a+\frac{709794461767354}{81467152395643}$, $\frac{106327271625979}{81467152395643}a^{20}-\frac{94887774139051}{162934304791286}a^{19}-\frac{65\cdots 25}{162934304791286}a^{18}+\frac{23\cdots 45}{162934304791286}a^{17}+\frac{11\cdots 81}{23276329255898}a^{16}-\frac{27\cdots 49}{162934304791286}a^{15}-\frac{45\cdots 11}{162934304791286}a^{14}+\frac{16\cdots 51}{162934304791286}a^{13}+\frac{10\cdots 68}{11638164627949}a^{12}-\frac{50\cdots 31}{162934304791286}a^{11}-\frac{13\cdots 58}{81467152395643}a^{10}+\frac{75\cdots 97}{162934304791286}a^{9}+\frac{28\cdots 45}{162934304791286}a^{8}-\frac{75\cdots 05}{23276329255898}a^{7}-\frac{16\cdots 01}{162934304791286}a^{6}+\frac{13\cdots 99}{162934304791286}a^{5}+\frac{23\cdots 92}{81467152395643}a^{4}-\frac{28\cdots 11}{81467152395643}a^{3}-\frac{59\cdots 73}{162934304791286}a^{2}-\frac{117014040664038}{11638164627949}a+\frac{298161726976627}{23276329255898}$, $\frac{375798294165759}{162934304791286}a^{20}-\frac{138836906847815}{81467152395643}a^{19}-\frac{57\cdots 52}{81467152395643}a^{18}+\frac{531869961684541}{11638164627949}a^{17}+\frac{13\cdots 53}{162934304791286}a^{16}-\frac{87\cdots 99}{162934304791286}a^{15}-\frac{39\cdots 77}{81467152395643}a^{14}+\frac{51\cdots 49}{162934304791286}a^{13}+\frac{25\cdots 19}{162934304791286}a^{12}-\frac{78\cdots 21}{81467152395643}a^{11}-\frac{22\cdots 00}{81467152395643}a^{10}+\frac{25\cdots 61}{162934304791286}a^{9}+\frac{23\cdots 14}{81467152395643}a^{8}-\frac{10\cdots 73}{81467152395643}a^{7}-\frac{28\cdots 89}{162934304791286}a^{6}+\frac{80\cdots 13}{162934304791286}a^{5}+\frac{85\cdots 93}{162934304791286}a^{4}-\frac{16\cdots 25}{23276329255898}a^{3}-\frac{11\cdots 59}{162934304791286}a^{2}+\frac{22\cdots 07}{162934304791286}a+\frac{19\cdots 27}{81467152395643}$, $\frac{263888766231187}{23276329255898}a^{20}-\frac{405791254392511}{81467152395643}a^{19}-\frac{28\cdots 81}{81467152395643}a^{18}+\frac{97\cdots 23}{81467152395643}a^{17}+\frac{33\cdots 32}{81467152395643}a^{16}-\frac{11\cdots 65}{81467152395643}a^{15}-\frac{19\cdots 65}{81467152395643}a^{14}+\frac{14\cdots 43}{162934304791286}a^{13}+\frac{12\cdots 17}{162934304791286}a^{12}-\frac{41\cdots 71}{162934304791286}a^{11}-\frac{21\cdots 39}{162934304791286}a^{10}+\frac{42\cdots 39}{11638164627949}a^{9}+\frac{10\cdots 82}{81467152395643}a^{8}-\frac{19\cdots 81}{81467152395643}a^{7}-\frac{56\cdots 49}{81467152395643}a^{6}+\frac{89\cdots 59}{162934304791286}a^{5}+\frac{13\cdots 37}{81467152395643}a^{4}-\frac{66\cdots 99}{162934304791286}a^{3}-\frac{22\cdots 23}{162934304791286}a^{2}+\frac{28\cdots 91}{162934304791286}a+\frac{58\cdots 03}{162934304791286}$, $\frac{181581521682513}{81467152395643}a^{20}-\frac{55956727239480}{81467152395643}a^{19}-\frac{11\cdots 23}{162934304791286}a^{18}+\frac{23\cdots 21}{162934304791286}a^{17}+\frac{65\cdots 35}{81467152395643}a^{16}-\frac{13\cdots 16}{81467152395643}a^{15}-\frac{38\cdots 74}{81467152395643}a^{14}+\frac{16\cdots 61}{162934304791286}a^{13}+\frac{24\cdots 85}{162934304791286}a^{12}-\frac{46\cdots 29}{162934304791286}a^{11}-\frac{22\cdots 06}{81467152395643}a^{10}+\frac{25\cdots 57}{81467152395643}a^{9}+\frac{44\cdots 53}{162934304791286}a^{8}-\frac{42\cdots 89}{81467152395643}a^{7}-\frac{23\cdots 69}{162934304791286}a^{6}-\frac{86\cdots 61}{81467152395643}a^{5}+\frac{35\cdots 27}{11638164627949}a^{4}+\frac{26\cdots 06}{81467152395643}a^{3}-\frac{50\cdots 89}{23276329255898}a^{2}-\frac{858223900150018}{81467152395643}a+\frac{429694553232411}{81467152395643}$, $\frac{95541883114478}{11638164627949}a^{20}-\frac{65511902286255}{23276329255898}a^{19}-\frac{41\cdots 01}{162934304791286}a^{18}+\frac{50\cdots 70}{81467152395643}a^{17}+\frac{48\cdots 15}{162934304791286}a^{16}-\frac{12\cdots 59}{162934304791286}a^{15}-\frac{39\cdots 75}{23276329255898}a^{14}+\frac{73\cdots 75}{162934304791286}a^{13}+\frac{88\cdots 09}{162934304791286}a^{12}-\frac{20\cdots 39}{162934304791286}a^{11}-\frac{22\cdots 69}{23276329255898}a^{10}+\frac{13\cdots 39}{81467152395643}a^{9}+\frac{80\cdots 76}{81467152395643}a^{8}-\frac{11\cdots 77}{162934304791286}a^{7}-\frac{83\cdots 71}{162934304791286}a^{6}-\frac{13\cdots 22}{11638164627949}a^{5}+\frac{18\cdots 01}{162934304791286}a^{4}+\frac{11\cdots 33}{162934304791286}a^{3}-\frac{75\cdots 59}{81467152395643}a^{2}-\frac{22\cdots 54}{81467152395643}a+\frac{35\cdots 23}{162934304791286}$, $\frac{131086708557461}{23276329255898}a^{20}-\frac{182153125809609}{81467152395643}a^{19}-\frac{14\cdots 23}{81467152395643}a^{18}+\frac{84\cdots 33}{162934304791286}a^{17}+\frac{32\cdots 29}{162934304791286}a^{16}-\frac{71\cdots 02}{11638164627949}a^{15}-\frac{95\cdots 08}{81467152395643}a^{14}+\frac{60\cdots 25}{162934304791286}a^{13}+\frac{30\cdots 49}{81467152395643}a^{12}-\frac{12\cdots 51}{11638164627949}a^{11}-\frac{53\cdots 62}{81467152395643}a^{10}+\frac{11\cdots 59}{81467152395643}a^{9}+\frac{53\cdots 26}{81467152395643}a^{8}-\frac{12\cdots 49}{162934304791286}a^{7}-\frac{39\cdots 04}{11638164627949}a^{6}+\frac{44\cdots 76}{81467152395643}a^{5}+\frac{62\cdots 89}{81467152395643}a^{4}+\frac{42\cdots 71}{162934304791286}a^{3}-\frac{98\cdots 09}{162934304791286}a^{2}-\frac{11\cdots 10}{81467152395643}a+\frac{337137305167777}{23276329255898}$, $\frac{310257168558130}{81467152395643}a^{20}-\frac{294420101635685}{162934304791286}a^{19}-\frac{19\cdots 41}{162934304791286}a^{18}+\frac{517488540458439}{11638164627949}a^{17}+\frac{22\cdots 55}{162934304791286}a^{16}-\frac{43\cdots 78}{81467152395643}a^{15}-\frac{12\cdots 23}{162934304791286}a^{14}+\frac{52\cdots 23}{162934304791286}a^{13}+\frac{41\cdots 19}{162934304791286}a^{12}-\frac{15\cdots 27}{162934304791286}a^{11}-\frac{37\cdots 19}{81467152395643}a^{10}+\frac{11\cdots 37}{81467152395643}a^{9}+\frac{75\cdots 81}{162934304791286}a^{8}-\frac{81\cdots 33}{81467152395643}a^{7}-\frac{20\cdots 86}{81467152395643}a^{6}+\frac{22\cdots 48}{81467152395643}a^{5}+\frac{96\cdots 97}{162934304791286}a^{4}-\frac{32\cdots 63}{11638164627949}a^{3}-\frac{87\cdots 29}{162934304791286}a^{2}+\frac{448145104718623}{162934304791286}a+\frac{12\cdots 98}{81467152395643}$, $\frac{14\cdots 89}{162934304791286}a^{20}-\frac{283214718416211}{81467152395643}a^{19}-\frac{21\cdots 81}{81467152395643}a^{18}+\frac{948457024480009}{11638164627949}a^{17}+\frac{25\cdots 72}{81467152395643}a^{16}-\frac{15\cdots 07}{162934304791286}a^{15}-\frac{14\cdots 78}{81467152395643}a^{14}+\frac{96\cdots 47}{162934304791286}a^{13}+\frac{92\cdots 13}{162934304791286}a^{12}-\frac{40\cdots 19}{23276329255898}a^{11}-\frac{83\cdots 98}{81467152395643}a^{10}+\frac{20\cdots 93}{81467152395643}a^{9}+\frac{16\cdots 81}{162934304791286}a^{8}-\frac{24\cdots 03}{162934304791286}a^{7}-\frac{88\cdots 43}{162934304791286}a^{6}+\frac{68\cdots 71}{23276329255898}a^{5}+\frac{10\cdots 55}{81467152395643}a^{4}-\frac{11\cdots 99}{11638164627949}a^{3}-\frac{12\cdots 19}{11638164627949}a^{2}+\frac{97987646202049}{11638164627949}a+\frac{20\cdots 94}{81467152395643}$, $\frac{359104450583044}{81467152395643}a^{20}-\frac{157519804217175}{81467152395643}a^{19}-\frac{22\cdots 21}{162934304791286}a^{18}+\frac{37\cdots 19}{81467152395643}a^{17}+\frac{25\cdots 39}{162934304791286}a^{16}-\frac{89\cdots 69}{162934304791286}a^{15}-\frac{75\cdots 17}{81467152395643}a^{14}+\frac{53\cdots 45}{162934304791286}a^{13}+\frac{23\cdots 61}{81467152395643}a^{12}-\frac{78\cdots 33}{81467152395643}a^{11}-\frac{43\cdots 95}{81467152395643}a^{10}+\frac{11\cdots 19}{81467152395643}a^{9}+\frac{43\cdots 27}{81467152395643}a^{8}-\frac{70\cdots 41}{81467152395643}a^{7}-\frac{23\cdots 22}{81467152395643}a^{6}+\frac{30\cdots 85}{162934304791286}a^{5}+\frac{80\cdots 83}{11638164627949}a^{4}-\frac{19\cdots 69}{162934304791286}a^{3}-\frac{51\cdots 49}{81467152395643}a^{2}+\frac{120198114333033}{11638164627949}a+\frac{24\cdots 75}{162934304791286}$, $\frac{413191994015709}{162934304791286}a^{20}+\frac{19623172707883}{162934304791286}a^{19}-\frac{63\cdots 34}{81467152395643}a^{18}-\frac{18\cdots 17}{162934304791286}a^{17}+\frac{14\cdots 77}{162934304791286}a^{16}+\frac{20\cdots 95}{162934304791286}a^{15}-\frac{85\cdots 71}{162934304791286}a^{14}-\frac{10\cdots 79}{162934304791286}a^{13}+\frac{27\cdots 95}{162934304791286}a^{12}+\frac{41\cdots 03}{162934304791286}a^{11}-\frac{24\cdots 44}{81467152395643}a^{10}-\frac{53\cdots 27}{81467152395643}a^{9}+\frac{47\cdots 05}{162934304791286}a^{8}+\frac{15\cdots 57}{162934304791286}a^{7}-\frac{11\cdots 06}{81467152395643}a^{6}-\frac{10\cdots 59}{162934304791286}a^{5}+\frac{31\cdots 11}{162934304791286}a^{4}+\frac{11\cdots 15}{81467152395643}a^{3}+\frac{972622189038192}{11638164627949}a^{2}-\frac{10\cdots 77}{162934304791286}a-\frac{798084424122007}{81467152395643}$, $\frac{840457270482821}{162934304791286}a^{20}-\frac{232669234685117}{162934304791286}a^{19}-\frac{25\cdots 33}{162934304791286}a^{18}+\frac{46\cdots 95}{162934304791286}a^{17}+\frac{43\cdots 85}{23276329255898}a^{16}-\frac{28\cdots 90}{81467152395643}a^{15}-\frac{17\cdots 63}{162934304791286}a^{14}+\frac{18\cdots 86}{81467152395643}a^{13}+\frac{80\cdots 11}{23276329255898}a^{12}-\frac{51\cdots 86}{81467152395643}a^{11}-\frac{51\cdots 07}{81467152395643}a^{10}+\frac{12\cdots 61}{162934304791286}a^{9}+\frac{51\cdots 12}{81467152395643}a^{8}-\frac{24\cdots 41}{11638164627949}a^{7}-\frac{27\cdots 43}{81467152395643}a^{6}-\frac{14\cdots 10}{81467152395643}a^{5}+\frac{62\cdots 07}{81467152395643}a^{4}+\frac{50\cdots 86}{81467152395643}a^{3}-\frac{10\cdots 35}{162934304791286}a^{2}-\frac{351753039908881}{11638164627949}a+\frac{364585407330227}{23276329255898}$, $\frac{31835304848954}{11638164627949}a^{20}-\frac{123418446432147}{162934304791286}a^{19}-\frac{68\cdots 41}{81467152395643}a^{18}+\frac{12\cdots 22}{81467152395643}a^{17}+\frac{79\cdots 58}{81467152395643}a^{16}-\frac{30\cdots 25}{162934304791286}a^{15}-\frac{91\cdots 37}{162934304791286}a^{14}+\frac{92\cdots 39}{81467152395643}a^{13}+\frac{28\cdots 41}{162934304791286}a^{12}-\frac{50\cdots 87}{162934304791286}a^{11}-\frac{25\cdots 24}{81467152395643}a^{10}+\frac{37\cdots 90}{11638164627949}a^{9}+\frac{49\cdots 51}{162934304791286}a^{8}+\frac{32\cdots 60}{81467152395643}a^{7}-\frac{12\cdots 81}{81467152395643}a^{6}-\frac{27\cdots 47}{162934304791286}a^{5}+\frac{48\cdots 75}{162934304791286}a^{4}+\frac{87\cdots 13}{162934304791286}a^{3}-\frac{26\cdots 03}{162934304791286}a^{2}-\frac{40\cdots 25}{162934304791286}a+\frac{343356653296589}{162934304791286}$, $\frac{362273560755107}{81467152395643}a^{20}-\frac{15844051852175}{11638164627949}a^{19}-\frac{22\cdots 03}{162934304791286}a^{18}+\frac{23\cdots 50}{81467152395643}a^{17}+\frac{26\cdots 81}{162934304791286}a^{16}-\frac{56\cdots 81}{162934304791286}a^{15}-\frac{10\cdots 93}{11638164627949}a^{14}+\frac{35\cdots 11}{162934304791286}a^{13}+\frac{24\cdots 45}{81467152395643}a^{12}-\frac{49\cdots 17}{81467152395643}a^{11}-\frac{43\cdots 79}{81467152395643}a^{10}+\frac{61\cdots 82}{81467152395643}a^{9}+\frac{43\cdots 42}{81467152395643}a^{8}-\frac{30\cdots 70}{11638164627949}a^{7}-\frac{22\cdots 86}{81467152395643}a^{6}-\frac{18\cdots 55}{162934304791286}a^{5}+\frac{51\cdots 00}{81467152395643}a^{4}+\frac{79\cdots 35}{162934304791286}a^{3}-\frac{41\cdots 39}{81467152395643}a^{2}-\frac{20\cdots 80}{81467152395643}a+\frac{21\cdots 85}{162934304791286}$ 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:  \( 1629233271.9 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 1629233271.9 \cdot 1}{2\cdot\sqrt{94142881806955162927406195366237}}\cr\approx \mathstrut & 0.17607165226 \end{aligned}\] (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^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*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)));
 
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^21 - 31*x^19 - 3*x^18 + 362*x^17 + 32*x^16 - 2119*x^15 - 158*x^14 + 6826*x^13 + 676*x^12 - 12509*x^11 - 2021*x^10 + 12809*x^9 + 3175*x^8 - 6710*x^7 - 2254*x^6 + 1442*x^5 + 571*x^4 - 97*x^3 - 44*x^2 + 2*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

$F_7$ (as 21T4):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.12431698517.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 42
Degree 7 sibling: 7.7.12431698517.1
Degree 14 sibling: 14.14.26736653147041339876997.1
Minimal sibling: 7.7.12431698517.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{3}{,}\,{\href{/padicField/2.3.0.1}{3} }$ ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ R ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.7.0.1}{7} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{7}$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{7}$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(7\) Copy content Toggle raw display 7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
\(173\) Copy content Toggle raw display 173.3.1.0a1.1$x^{3} + 2 x + 171$$1$$3$$0$$C_3$$$[\ ]^{3}$$
173.3.2.3a1.2$x^{6} + 4 x^{4} + 342 x^{3} + 4 x^{2} + 684 x + 29414$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
173.3.2.3a1.2$x^{6} + 4 x^{4} + 342 x^{3} + 4 x^{2} + 684 x + 29414$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
173.3.2.3a1.2$x^{6} + 4 x^{4} + 342 x^{3} + 4 x^{2} + 684 x + 29414$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

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