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Properties

Label 18.18.200...136.2
Degree $18$
Signature $[18, 0]$
Discriminant $2.008\times 10^{36}$
Root discriminant \(103.95\)
Ramified primes $2,3,7,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4^3:(C_2\times A_4)$ (as 18T701)

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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^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321)
 
Copy content gp:K = bnfinit(y^18 - 105*y^16 + 4536*y^14 - 105014*y^12 + 1420839*y^10 - 11417007*y^8 + 52263162*y^6 - 119427945*y^4 + 92681883*y^2 - 6964321, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321)
 

\( x^{18} - 105 x^{16} + 4536 x^{14} - 105014 x^{12} + 1420839 x^{10} - 11417007 x^{8} + 52263162 x^{6} + \cdots - 6964321 \) 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:  $18$
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:  $[18, 0]$
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:   \(2007609002750174051598743153244635136\) \(\medspace = 2^{18}\cdot 3^{18}\cdot 7^{16}\cdot 29^{6}\) 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:  \(103.95\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{63/32}3^{25/18}7^{11/12}29^{5/6}\approx 1772.7986604582713$
Ramified primes:   \(2\), \(3\), \(7\), \(29\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}$, $\frac{1}{7}a^{13}$, $\frac{1}{7}a^{14}$, $\frac{1}{7}a^{15}$, $\frac{1}{18\cdots 91}a^{16}+\frac{67\cdots 79}{18\cdots 91}a^{14}+\frac{14\cdots 80}{18\cdots 91}a^{12}-\frac{41\cdots 59}{26\cdots 13}a^{10}+\frac{83\cdots 41}{26\cdots 13}a^{8}-\frac{96\cdots 64}{26\cdots 13}a^{6}-\frac{45\cdots 26}{90\cdots 97}a^{4}+\frac{54\cdots 76}{90\cdots 97}a^{2}+\frac{11\cdots 56}{90\cdots 97}$, $\frac{1}{23\cdots 83}a^{17}+\frac{13\cdots 44}{23\cdots 83}a^{15}+\frac{10\cdots 32}{23\cdots 83}a^{13}-\frac{31\cdots 43}{26\cdots 13}a^{11}+\frac{16\cdots 19}{34\cdots 69}a^{9}-\frac{16\cdots 42}{34\cdots 69}a^{7}-\frac{49\cdots 11}{11\cdots 61}a^{5}+\frac{21\cdots 59}{90\cdots 97}a^{3}+\frac{11\cdots 56}{11\cdots 61}a$ 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:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (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:  $17$
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:   $\frac{26\cdots 08}{42\cdots 27}a^{16}+\frac{25\cdots 70}{42\cdots 27}a^{14}-\frac{13\cdots 09}{60\cdots 61}a^{12}+\frac{25\cdots 47}{60\cdots 61}a^{10}-\frac{26\cdots 17}{60\cdots 61}a^{8}+\frac{15\cdots 35}{60\cdots 61}a^{6}-\frac{14\cdots 61}{20\cdots 09}a^{4}+\frac{11\cdots 78}{20\cdots 09}a^{2}-\frac{41\cdots 95}{20\cdots 09}$, $\frac{26\cdots 08}{42\cdots 27}a^{16}-\frac{25\cdots 70}{42\cdots 27}a^{14}+\frac{13\cdots 09}{60\cdots 61}a^{12}-\frac{25\cdots 47}{60\cdots 61}a^{10}+\frac{26\cdots 17}{60\cdots 61}a^{8}-\frac{15\cdots 35}{60\cdots 61}a^{6}+\frac{14\cdots 61}{20\cdots 09}a^{4}-\frac{11\cdots 78}{20\cdots 09}a^{2}+\frac{62\cdots 04}{20\cdots 09}$, $\frac{82\cdots 98}{18\cdots 91}a^{16}-\frac{74\cdots 12}{18\cdots 91}a^{14}+\frac{26\cdots 43}{18\cdots 91}a^{12}-\frac{66\cdots 96}{26\cdots 13}a^{10}+\frac{63\cdots 93}{26\cdots 13}a^{8}-\frac{32\cdots 45}{26\cdots 13}a^{6}+\frac{27\cdots 03}{90\cdots 97}a^{4}-\frac{26\cdots 71}{90\cdots 97}a^{2}+\frac{65\cdots 27}{90\cdots 97}$, $\frac{22\cdots 27}{63\cdots 79}a^{16}-\frac{20\cdots 06}{63\cdots 79}a^{14}+\frac{76\cdots 33}{63\cdots 79}a^{12}-\frac{20\cdots 90}{90\cdots 97}a^{10}+\frac{20\cdots 89}{90\cdots 97}a^{8}-\frac{11\cdots 09}{90\cdots 97}a^{6}+\frac{29\cdots 41}{90\cdots 97}a^{4}-\frac{24\cdots 24}{90\cdots 97}a^{2}+\frac{20\cdots 61}{90\cdots 97}$, $\frac{14\cdots 60}{18\cdots 91}a^{16}+\frac{13\cdots 29}{18\cdots 91}a^{14}-\frac{54\cdots 68}{18\cdots 91}a^{12}+\frac{15\cdots 60}{26\cdots 13}a^{10}-\frac{17\cdots 53}{26\cdots 13}a^{8}+\frac{10\cdots 00}{26\cdots 13}a^{6}-\frac{10\cdots 28}{90\cdots 97}a^{4}+\frac{97\cdots 55}{90\cdots 97}a^{2}-\frac{73\cdots 08}{90\cdots 97}$, $\frac{32\cdots 93}{26\cdots 13}a^{16}+\frac{22\cdots 65}{18\cdots 91}a^{14}-\frac{91\cdots 57}{18\cdots 91}a^{12}+\frac{26\cdots 68}{26\cdots 13}a^{10}-\frac{30\cdots 07}{26\cdots 13}a^{8}+\frac{18\cdots 01}{26\cdots 13}a^{6}-\frac{18\cdots 24}{90\cdots 97}a^{4}+\frac{16\cdots 29}{90\cdots 97}a^{2}-\frac{12\cdots 70}{90\cdots 97}$, $\frac{51\cdots 09}{26\cdots 13}a^{16}-\frac{48\cdots 68}{26\cdots 13}a^{14}+\frac{18\cdots 15}{26\cdots 13}a^{12}-\frac{35\cdots 33}{26\cdots 13}a^{10}+\frac{37\cdots 81}{26\cdots 13}a^{8}-\frac{21\cdots 98}{26\cdots 13}a^{6}+\frac{19\cdots 75}{90\cdots 97}a^{4}-\frac{16\cdots 86}{90\cdots 97}a^{2}+\frac{12\cdots 14}{90\cdots 97}$, $\frac{27\cdots 52}{26\cdots 13}a^{16}-\frac{17\cdots 78}{18\cdots 91}a^{14}+\frac{60\cdots 35}{18\cdots 91}a^{12}-\frac{15\cdots 96}{26\cdots 13}a^{10}+\frac{14\cdots 01}{26\cdots 13}a^{8}-\frac{76\cdots 67}{26\cdots 13}a^{6}+\frac{67\cdots 16}{90\cdots 97}a^{4}-\frac{63\cdots 99}{90\cdots 97}a^{2}+\frac{50\cdots 39}{90\cdots 97}$, $\frac{89\cdots 42}{23\cdots 83}a^{17}+\frac{39\cdots 02}{26\cdots 13}a^{16}-\frac{12\cdots 28}{34\cdots 69}a^{15}-\frac{24\cdots 19}{18\cdots 91}a^{14}+\frac{31\cdots 25}{23\cdots 83}a^{13}+\frac{84\cdots 78}{18\cdots 91}a^{12}-\frac{67\cdots 35}{26\cdots 13}a^{11}-\frac{20\cdots 87}{26\cdots 13}a^{10}+\frac{95\cdots 76}{34\cdots 69}a^{9}+\frac{18\cdots 68}{26\cdots 13}a^{8}-\frac{57\cdots 68}{34\cdots 69}a^{7}-\frac{79\cdots 97}{26\cdots 13}a^{6}+\frac{63\cdots 40}{11\cdots 61}a^{5}+\frac{37\cdots 04}{90\cdots 97}a^{4}-\frac{70\cdots 53}{90\cdots 97}a^{3}+\frac{61\cdots 88}{90\cdots 97}a^{2}+\frac{47\cdots 67}{11\cdots 61}a-\frac{92\cdots 11}{90\cdots 97}$, $\frac{88\cdots 78}{23\cdots 83}a^{17}-\frac{14\cdots 95}{18\cdots 91}a^{16}-\frac{83\cdots 68}{23\cdots 83}a^{15}+\frac{13\cdots 54}{18\cdots 91}a^{14}+\frac{31\cdots 58}{23\cdots 83}a^{13}-\frac{48\cdots 97}{18\cdots 91}a^{12}-\frac{66\cdots 55}{26\cdots 13}a^{11}+\frac{12\cdots 26}{26\cdots 13}a^{10}+\frac{91\cdots 63}{34\cdots 69}a^{9}-\frac{12\cdots 20}{26\cdots 13}a^{8}-\frac{53\cdots 33}{34\cdots 69}a^{7}+\frac{66\cdots 95}{26\cdots 13}a^{6}+\frac{55\cdots 93}{11\cdots 61}a^{5}-\frac{56\cdots 41}{90\cdots 97}a^{4}-\frac{51\cdots 06}{90\cdots 97}a^{3}+\frac{40\cdots 26}{90\cdots 97}a^{2}+\frac{18\cdots 68}{11\cdots 61}a-\frac{59\cdots 05}{90\cdots 97}$, $\frac{29\cdots 19}{23\cdots 83}a^{17}+\frac{10\cdots 31}{26\cdots 13}a^{16}-\frac{27\cdots 05}{23\cdots 83}a^{15}-\frac{88\cdots 94}{18\cdots 91}a^{14}+\frac{10\cdots 66}{23\cdots 83}a^{13}+\frac{41\cdots 81}{18\cdots 91}a^{12}-\frac{20\cdots 65}{26\cdots 13}a^{11}-\frac{14\cdots 70}{26\cdots 13}a^{10}+\frac{26\cdots 14}{34\cdots 69}a^{9}+\frac{17\cdots 09}{26\cdots 13}a^{8}-\frac{14\cdots 37}{34\cdots 69}a^{7}-\frac{11\cdots 70}{26\cdots 13}a^{6}+\frac{13\cdots 99}{11\cdots 61}a^{5}+\frac{98\cdots 81}{90\cdots 97}a^{4}-\frac{86\cdots 32}{90\cdots 97}a^{3}-\frac{23\cdots 51}{90\cdots 97}a^{2}+\frac{63\cdots 49}{11\cdots 61}a+\frac{56\cdots 47}{90\cdots 97}$, $\frac{14\cdots 00}{23\cdots 83}a^{17}-\frac{32\cdots 32}{18\cdots 91}a^{16}-\frac{21\cdots 76}{34\cdots 69}a^{15}+\frac{31\cdots 98}{18\cdots 91}a^{14}+\frac{87\cdots 89}{34\cdots 69}a^{13}-\frac{16\cdots 32}{26\cdots 13}a^{12}-\frac{13\cdots 48}{26\cdots 13}a^{11}+\frac{33\cdots 97}{26\cdots 13}a^{10}+\frac{19\cdots 23}{34\cdots 69}a^{9}-\frac{36\cdots 14}{26\cdots 13}a^{8}-\frac{10\cdots 53}{34\cdots 69}a^{7}+\frac{21\cdots 69}{26\cdots 13}a^{6}+\frac{70\cdots 03}{11\cdots 61}a^{5}-\frac{20\cdots 02}{90\cdots 97}a^{4}+\frac{60\cdots 18}{90\cdots 97}a^{3}+\frac{17\cdots 65}{90\cdots 97}a^{2}-\frac{12\cdots 18}{11\cdots 61}a-\frac{13\cdots 50}{90\cdots 97}$, $\frac{29\cdots 45}{23\cdots 83}a^{17}+\frac{19\cdots 62}{18\cdots 91}a^{16}+\frac{27\cdots 90}{23\cdots 83}a^{15}-\frac{17\cdots 07}{18\cdots 91}a^{14}-\frac{10\cdots 21}{23\cdots 83}a^{13}+\frac{61\cdots 09}{18\cdots 91}a^{12}+\frac{21\cdots 20}{26\cdots 13}a^{11}-\frac{14\cdots 53}{26\cdots 13}a^{10}-\frac{29\cdots 42}{34\cdots 69}a^{9}+\frac{13\cdots 86}{26\cdots 13}a^{8}+\frac{17\cdots 30}{34\cdots 69}a^{7}-\frac{59\cdots 69}{26\cdots 13}a^{6}-\frac{19\cdots 70}{11\cdots 61}a^{5}+\frac{34\cdots 60}{90\cdots 97}a^{4}+\frac{23\cdots 19}{90\cdots 97}a^{3}+\frac{79\cdots 03}{90\cdots 97}a^{2}-\frac{17\cdots 77}{11\cdots 61}a-\frac{34\cdots 49}{90\cdots 97}$, $\frac{22\cdots 20}{23\cdots 83}a^{17}+\frac{14\cdots 91}{18\cdots 91}a^{16}-\frac{20\cdots 42}{23\cdots 83}a^{15}-\frac{13\cdots 97}{18\cdots 91}a^{14}+\frac{74\cdots 18}{23\cdots 83}a^{13}+\frac{50\cdots 13}{18\cdots 91}a^{12}-\frac{14\cdots 30}{26\cdots 13}a^{11}-\frac{13\cdots 51}{26\cdots 13}a^{10}+\frac{18\cdots 93}{34\cdots 69}a^{9}+\frac{14\cdots 96}{26\cdots 13}a^{8}-\frac{96\cdots 31}{34\cdots 69}a^{7}-\frac{81\cdots 96}{26\cdots 13}a^{6}+\frac{77\cdots 71}{11\cdots 61}a^{5}+\frac{75\cdots 19}{90\cdots 97}a^{4}-\frac{33\cdots 50}{90\cdots 97}a^{3}-\frac{62\cdots 76}{90\cdots 97}a^{2}-\frac{16\cdots 88}{11\cdots 61}a+\frac{43\cdots 30}{90\cdots 97}$, $\frac{49\cdots 09}{23\cdots 83}a^{17}+\frac{10\cdots 42}{18\cdots 91}a^{16}-\frac{47\cdots 71}{23\cdots 83}a^{15}-\frac{96\cdots 09}{18\cdots 91}a^{14}+\frac{17\cdots 90}{23\cdots 83}a^{13}+\frac{36\cdots 66}{18\cdots 91}a^{12}-\frac{37\cdots 50}{26\cdots 13}a^{11}-\frac{99\cdots 34}{26\cdots 13}a^{10}+\frac{51\cdots 74}{34\cdots 69}a^{9}+\frac{10\cdots 61}{26\cdots 13}a^{8}-\frac{28\cdots 37}{34\cdots 69}a^{7}-\frac{59\cdots 25}{26\cdots 13}a^{6}+\frac{25\cdots 59}{11\cdots 61}a^{5}+\frac{55\cdots 59}{90\cdots 97}a^{4}-\frac{13\cdots 63}{90\cdots 97}a^{3}-\frac{48\cdots 31}{90\cdots 97}a^{2}+\frac{60\cdots 07}{11\cdots 61}a+\frac{37\cdots 64}{90\cdots 97}$, $\frac{30\cdots 07}{18\cdots 91}a^{17}-\frac{68\cdots 98}{18\cdots 91}a^{16}+\frac{39\cdots 18}{26\cdots 13}a^{15}+\frac{66\cdots 26}{18\cdots 91}a^{14}-\frac{14\cdots 00}{26\cdots 13}a^{13}-\frac{25\cdots 19}{18\cdots 91}a^{12}+\frac{26\cdots 78}{26\cdots 13}a^{11}+\frac{73\cdots 87}{26\cdots 13}a^{10}-\frac{26\cdots 15}{26\cdots 13}a^{9}-\frac{79\cdots 64}{26\cdots 13}a^{8}+\frac{13\cdots 12}{26\cdots 13}a^{7}+\frac{46\cdots 63}{26\cdots 13}a^{6}-\frac{11\cdots 26}{90\cdots 97}a^{5}-\frac{44\cdots 54}{90\cdots 97}a^{4}+\frac{84\cdots 70}{90\cdots 97}a^{3}+\frac{38\cdots 78}{90\cdots 97}a^{2}-\frac{66\cdots 39}{90\cdots 97}a-\frac{28\cdots 83}{90\cdots 97}$, $\frac{50\cdots 44}{23\cdots 83}a^{17}+\frac{80\cdots 80}{18\cdots 91}a^{16}+\frac{68\cdots 02}{34\cdots 69}a^{15}-\frac{77\cdots 73}{18\cdots 91}a^{14}-\frac{17\cdots 47}{23\cdots 83}a^{13}+\frac{29\cdots 76}{18\cdots 91}a^{12}+\frac{38\cdots 89}{26\cdots 13}a^{11}-\frac{81\cdots 48}{26\cdots 13}a^{10}-\frac{52\cdots 68}{34\cdots 69}a^{9}+\frac{86\cdots 87}{26\cdots 13}a^{8}+\frac{29\cdots 89}{34\cdots 69}a^{7}-\frac{49\cdots 52}{26\cdots 13}a^{6}-\frac{27\cdots 49}{11\cdots 61}a^{5}+\frac{46\cdots 24}{90\cdots 97}a^{4}+\frac{18\cdots 40}{90\cdots 97}a^{3}-\frac{40\cdots 95}{90\cdots 97}a^{2}-\frac{17\cdots 65}{11\cdots 61}a+\frac{34\cdots 73}{90\cdots 97}$ 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:  \( 2820691242954.3164 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 2820691242954.3164 \cdot 1}{2\cdot\sqrt{2007609002750174051598743153244635136}}\cr\approx \mathstrut & 0.260931138540750 \end{aligned}\] (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^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321) 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^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321, 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^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 105*x^16 + 4536*x^14 - 105014*x^12 + 1420839*x^10 - 11417007*x^8 + 52263162*x^6 - 119427945*x^4 + 92681883*x^2 - 6964321); 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

$A_4^3:(C_2\times A_4)$ (as 18T701):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 41472
The 72 conjugacy class representatives for $A_4^3:(C_2\times A_4)$
Character table for $A_4^3:(C_2\times A_4)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 R R ${\href{/padicField/5.9.0.1}{9} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ R ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.9.0.1}{9} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 2.3.2.6a1.1$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$$2$$3$$6$$C_6$$$[2]^{3}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(3\) Copy content Toggle raw display 3.3.3.9a8.1$x^{9} + 6 x^{7} + 3 x^{6} + 15 x^{5} + 12 x^{4} + 20 x^{3} + 15 x^{2} + 12 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
3.3.3.9a8.1$x^{9} + 6 x^{7} + 3 x^{6} + 15 x^{5} + 12 x^{4} + 20 x^{3} + 15 x^{2} + 12 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.6$x^{6} + 42$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.12.11a1.1$x^{12} + 7$$12$$1$$11$$D_4 \times C_3$$$[\ ]_{12}^{2}$$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{29}$$x + 27$$1$$1$$0$Trivial$$[\ ]$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
29.1.6.5a1.1$x^{6} + 29$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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