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Properties

Label 20.0.243...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $2.435\times 10^{39}$
Root discriminant \(93.18\)
Ramified primes $2,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.C_3^4:(S_3\times F_5)$ (as 20T1032)

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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^20 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180)
 
Copy content gp:K = bnfinit(y^20 - 20*y^18 + 220*y^16 - 48*y^15 - 520*y^14 + 1040*y^13 - 6670*y^12 - 8680*y^11 + 82648*y^10 + 80240*y^9 - 335920*y^8 - 172960*y^7 + 1105960*y^6 + 399184*y^5 - 1904775*y^4 + 1283240*y^3 + 8424540*y^2 + 8752080*y + 3070180, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180)
 

\( x^{20} - 20 x^{18} + 220 x^{16} - 48 x^{15} - 520 x^{14} + 1040 x^{13} - 6670 x^{12} - 8680 x^{11} + \cdots + 3070180 \) 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:   \(2434814898518097920000000000000000000000\) \(\medspace = 2^{59}\cdot 5^{22}\cdot 11^{6}\) 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:  \(93.18\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{8}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{9}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{8}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{20\cdots 52}a^{19}+\frac{25\cdots 43}{10\cdots 76}a^{18}-\frac{25\cdots 87}{51\cdots 38}a^{17}-\frac{55\cdots 29}{51\cdots 38}a^{16}+\frac{71\cdots 27}{51\cdots 38}a^{15}+\frac{38\cdots 89}{51\cdots 38}a^{14}-\frac{46\cdots 83}{51\cdots 38}a^{13}-\frac{39\cdots 25}{51\cdots 38}a^{12}+\frac{47\cdots 49}{10\cdots 76}a^{11}+\frac{33\cdots 23}{51\cdots 38}a^{10}-\frac{28\cdots 65}{10\cdots 76}a^{9}-\frac{11\cdots 60}{25\cdots 69}a^{8}+\frac{87\cdots 29}{51\cdots 38}a^{7}+\frac{10\cdots 33}{51\cdots 38}a^{6}-\frac{77\cdots 19}{25\cdots 69}a^{5}+\frac{11\cdots 43}{51\cdots 38}a^{4}+\frac{62\cdots 21}{20\cdots 52}a^{3}-\frac{41\cdots 21}{10\cdots 76}a^{2}-\frac{26\cdots 77}{10\cdots 76}a-\frac{92\cdots 13}{51\cdots 38}$ 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:  $9$
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:   \( -1 \)  (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:   $\frac{15\cdots 87}{51\cdots 38}a^{19}-\frac{97\cdots 21}{51\cdots 38}a^{18}-\frac{15\cdots 13}{25\cdots 69}a^{17}+\frac{10\cdots 19}{25\cdots 69}a^{16}+\frac{17\cdots 43}{25\cdots 69}a^{15}-\frac{15\cdots 09}{25\cdots 69}a^{14}-\frac{35\cdots 21}{25\cdots 69}a^{13}+\frac{46\cdots 25}{10\cdots 76}a^{12}-\frac{11\cdots 51}{51\cdots 38}a^{11}-\frac{68\cdots 65}{51\cdots 38}a^{10}+\frac{69\cdots 29}{25\cdots 69}a^{9}+\frac{34\cdots 63}{51\cdots 38}a^{8}-\frac{29\cdots 96}{25\cdots 69}a^{7}+\frac{69\cdots 96}{25\cdots 69}a^{6}+\frac{89\cdots 47}{25\cdots 69}a^{5}-\frac{14\cdots 85}{10\cdots 76}a^{4}-\frac{14\cdots 98}{25\cdots 69}a^{3}+\frac{21\cdots 97}{25\cdots 69}a^{2}+\frac{52\cdots 52}{25\cdots 69}a+\frac{27\cdots 36}{25\cdots 69}$, $\frac{90\cdots 03}{51\cdots 38}a^{19}-\frac{37\cdots 07}{51\cdots 38}a^{18}-\frac{18\cdots 55}{51\cdots 38}a^{17}+\frac{17\cdots 51}{10\cdots 76}a^{16}+\frac{20\cdots 95}{51\cdots 38}a^{15}-\frac{14\cdots 73}{51\cdots 38}a^{14}-\frac{49\cdots 65}{51\cdots 38}a^{13}+\frac{26\cdots 17}{10\cdots 76}a^{12}-\frac{66\cdots 69}{51\cdots 38}a^{11}-\frac{56\cdots 17}{51\cdots 38}a^{10}+\frac{81\cdots 73}{51\cdots 38}a^{9}+\frac{69\cdots 47}{10\cdots 76}a^{8}-\frac{35\cdots 45}{51\cdots 38}a^{7}+\frac{29\cdots 83}{51\cdots 38}a^{6}+\frac{11\cdots 27}{51\cdots 38}a^{5}-\frac{63\cdots 99}{10\cdots 76}a^{4}-\frac{93\cdots 06}{25\cdots 69}a^{3}+\frac{12\cdots 61}{25\cdots 69}a^{2}+\frac{33\cdots 86}{25\cdots 69}a+\frac{18\cdots 46}{25\cdots 69}$, $\frac{47\cdots 95}{51\cdots 38}a^{19}-\frac{30\cdots 03}{25\cdots 69}a^{18}-\frac{18\cdots 53}{10\cdots 76}a^{17}+\frac{23\cdots 35}{10\cdots 76}a^{16}+\frac{94\cdots 37}{51\cdots 38}a^{15}-\frac{30\cdots 89}{10\cdots 76}a^{14}-\frac{47\cdots 06}{25\cdots 69}a^{13}+\frac{14\cdots 35}{10\cdots 76}a^{12}-\frac{82\cdots 41}{10\cdots 76}a^{11}+\frac{15\cdots 07}{10\cdots 76}a^{10}+\frac{81\cdots 87}{10\cdots 76}a^{9}-\frac{31\cdots 31}{10\cdots 76}a^{8}-\frac{79\cdots 41}{25\cdots 69}a^{7}+\frac{28\cdots 59}{10\cdots 76}a^{6}+\frac{40\cdots 63}{51\cdots 38}a^{5}-\frac{88\cdots 29}{10\cdots 76}a^{4}-\frac{91\cdots 73}{10\cdots 76}a^{3}+\frac{29\cdots 71}{10\cdots 76}a^{2}+\frac{21\cdots 91}{51\cdots 38}a+\frac{75\cdots 21}{51\cdots 38}$, $\frac{48\cdots 36}{25\cdots 69}a^{19}-\frac{55\cdots 15}{10\cdots 76}a^{18}+\frac{15\cdots 65}{10\cdots 76}a^{17}+\frac{57\cdots 79}{10\cdots 76}a^{16}-\frac{11\cdots 79}{51\cdots 38}a^{15}-\frac{43\cdots 39}{10\cdots 76}a^{14}+\frac{25\cdots 59}{10\cdots 76}a^{13}-\frac{51\cdots 17}{10\cdots 76}a^{12}+\frac{95\cdots 38}{25\cdots 69}a^{11}+\frac{86\cdots 16}{25\cdots 69}a^{10}-\frac{83\cdots 07}{10\cdots 76}a^{9}-\frac{19\cdots 87}{10\cdots 76}a^{8}+\frac{28\cdots 17}{51\cdots 38}a^{7}+\frac{18\cdots 27}{10\cdots 76}a^{6}-\frac{16\cdots 37}{10\cdots 76}a^{5}+\frac{11\cdots 07}{10\cdots 76}a^{4}+\frac{73\cdots 32}{25\cdots 69}a^{3}-\frac{20\cdots 95}{10\cdots 76}a^{2}-\frac{18\cdots 46}{25\cdots 69}a-\frac{20\cdots 93}{51\cdots 38}$, $\frac{54\cdots 55}{10\cdots 76}a^{19}-\frac{50\cdots 24}{25\cdots 69}a^{18}-\frac{30\cdots 68}{25\cdots 69}a^{17}+\frac{23\cdots 89}{51\cdots 38}a^{16}+\frac{10\cdots 17}{10\cdots 76}a^{15}-\frac{56\cdots 57}{10\cdots 76}a^{14}+\frac{89\cdots 21}{51\cdots 38}a^{13}+\frac{25\cdots 61}{10\cdots 76}a^{12}-\frac{10\cdots 45}{10\cdots 76}a^{11}+\frac{25\cdots 89}{25\cdots 69}a^{10}+\frac{15\cdots 68}{25\cdots 69}a^{9}-\frac{45\cdots 09}{25\cdots 69}a^{8}-\frac{13\cdots 59}{10\cdots 76}a^{7}+\frac{96\cdots 53}{10\cdots 76}a^{6}-\frac{56\cdots 03}{51\cdots 38}a^{5}-\frac{42\cdots 61}{10\cdots 76}a^{4}-\frac{75\cdots 61}{25\cdots 69}a^{3}+\frac{26\cdots 41}{51\cdots 38}a^{2}+\frac{20\cdots 80}{25\cdots 69}a+\frac{83\cdots 73}{25\cdots 69}$, $\frac{37\cdots 87}{51\cdots 38}a^{19}-\frac{12\cdots 75}{10\cdots 76}a^{18}-\frac{13\cdots 67}{10\cdots 76}a^{17}+\frac{23\cdots 73}{10\cdots 76}a^{16}+\frac{11\cdots 09}{10\cdots 76}a^{15}-\frac{14\cdots 13}{51\cdots 38}a^{14}+\frac{20\cdots 29}{10\cdots 76}a^{13}+\frac{92\cdots 41}{10\cdots 76}a^{12}-\frac{76\cdots 93}{10\cdots 76}a^{11}+\frac{69\cdots 85}{10\cdots 76}a^{10}+\frac{47\cdots 75}{10\cdots 76}a^{9}-\frac{57\cdots 75}{10\cdots 76}a^{8}-\frac{12\cdots 55}{10\cdots 76}a^{7}+\frac{57\cdots 08}{25\cdots 69}a^{6}+\frac{24\cdots 25}{10\cdots 76}a^{5}-\frac{52\cdots 55}{10\cdots 76}a^{4}-\frac{20\cdots 39}{10\cdots 76}a^{3}+\frac{67\cdots 85}{51\cdots 38}a^{2}+\frac{81\cdots 81}{51\cdots 38}a+\frac{13\cdots 03}{25\cdots 69}$, $\frac{76\cdots 21}{10\cdots 76}a^{19}-\frac{28\cdots 21}{51\cdots 38}a^{18}-\frac{37\cdots 75}{25\cdots 69}a^{17}+\frac{33\cdots 38}{25\cdots 69}a^{16}+\frac{81\cdots 23}{51\cdots 38}a^{15}-\frac{19\cdots 51}{10\cdots 76}a^{14}-\frac{30\cdots 07}{10\cdots 76}a^{13}+\frac{13\cdots 09}{10\cdots 76}a^{12}-\frac{62\cdots 51}{10\cdots 76}a^{11}-\frac{51\cdots 29}{25\cdots 69}a^{10}+\frac{34\cdots 15}{51\cdots 38}a^{9}-\frac{25\cdots 81}{51\cdots 38}a^{8}-\frac{71\cdots 63}{25\cdots 69}a^{7}+\frac{17\cdots 35}{10\cdots 76}a^{6}+\frac{78\cdots 23}{10\cdots 76}a^{5}-\frac{60\cdots 01}{10\cdots 76}a^{4}-\frac{51\cdots 59}{51\cdots 38}a^{3}+\frac{60\cdots 88}{25\cdots 69}a^{2}+\frac{10\cdots 08}{25\cdots 69}a+\frac{44\cdots 24}{25\cdots 69}$, $\frac{22\cdots 51}{25\cdots 69}a^{19}-\frac{37\cdots 21}{51\cdots 38}a^{18}-\frac{17\cdots 47}{10\cdots 76}a^{17}+\frac{39\cdots 17}{25\cdots 69}a^{16}+\frac{18\cdots 41}{10\cdots 76}a^{15}-\frac{54\cdots 97}{25\cdots 69}a^{14}-\frac{32\cdots 65}{10\cdots 76}a^{13}+\frac{13\cdots 85}{10\cdots 76}a^{12}-\frac{71\cdots 43}{10\cdots 76}a^{11}-\frac{20\cdots 15}{10\cdots 76}a^{10}+\frac{78\cdots 31}{10\cdots 76}a^{9}+\frac{15\cdots 16}{25\cdots 69}a^{8}-\frac{32\cdots 43}{10\cdots 76}a^{7}+\frac{79\cdots 71}{51\cdots 38}a^{6}+\frac{92\cdots 87}{10\cdots 76}a^{5}-\frac{61\cdots 37}{10\cdots 76}a^{4}-\frac{13\cdots 31}{10\cdots 76}a^{3}+\frac{26\cdots 49}{10\cdots 76}a^{2}+\frac{25\cdots 35}{51\cdots 38}a+\frac{11\cdots 51}{51\cdots 38}$, $\frac{11\cdots 29}{10\cdots 76}a^{19}-\frac{48\cdots 10}{25\cdots 69}a^{18}-\frac{50\cdots 53}{25\cdots 69}a^{17}+\frac{35\cdots 85}{10\cdots 76}a^{16}+\frac{10\cdots 01}{51\cdots 38}a^{15}-\frac{10\cdots 58}{25\cdots 69}a^{14}+\frac{40\cdots 41}{10\cdots 76}a^{13}+\frac{14\cdots 29}{10\cdots 76}a^{12}-\frac{10\cdots 91}{10\cdots 76}a^{11}+\frac{31\cdots 39}{51\cdots 38}a^{10}+\frac{44\cdots 27}{51\cdots 38}a^{9}-\frac{67\cdots 07}{10\cdots 76}a^{8}-\frac{76\cdots 74}{25\cdots 69}a^{7}+\frac{10\cdots 05}{25\cdots 69}a^{6}+\frac{59\cdots 27}{10\cdots 76}a^{5}-\frac{12\cdots 67}{10\cdots 76}a^{4}-\frac{30\cdots 49}{51\cdots 38}a^{3}+\frac{13\cdots 05}{51\cdots 38}a^{2}+\frac{62\cdots 52}{25\cdots 69}a+\frac{74\cdots 89}{25\cdots 69}$ 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:  \( 962178480689 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 962178480689 \cdot 1}{2\cdot\sqrt{2434814898518097920000000000000000000000}}\cr\approx \mathstrut & 0.934956385888517 \end{aligned}\] (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^20 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180) 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 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180, 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 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180); 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 - 20*x^18 + 220*x^16 - 48*x^15 - 520*x^14 + 1040*x^13 - 6670*x^12 - 8680*x^11 + 82648*x^10 + 80240*x^9 - 335920*x^8 - 172960*x^7 + 1105960*x^6 + 399184*x^5 - 1904775*x^4 + 1283240*x^3 + 8424540*x^2 + 8752080*x + 3070180); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^{10}.C_3^4:(S_3\times F_5)$ (as 20T1032):

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 9953280
The 124 conjugacy class representatives for $C_2^{10}.C_3^4:(S_3\times F_5)$
Character table for $C_2^{10}.C_3^4:(S_3\times F_5)$

Intermediate fields

5.1.200000.1

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

Degree 30 siblings: data not computed
Degree 40 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ $20$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ $15{,}\,{\href{/padicField/31.5.0.1}{5} }$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }$ $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.1.4.11a1.2$x^{4} + 18$$4$$1$$11$$D_{4}$$$[2, 3, 4]$$
2.2.8.48b4.32$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 270 x^{12} + 528 x^{11} + 868 x^{10} + 1216 x^{9} + 1475 x^{8} + 1560 x^{7} + 1460 x^{6} + 1216 x^{5} + 898 x^{4} + 576 x^{3} + 304 x^{2} + 128 x + 31$$8$$2$$48$16T1385not computed
\(5\) Copy content Toggle raw display 5.1.5.5a1.4$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$$[\frac{5}{4}]_{4}$$
5.1.15.17a1.4$x^{15} + 20 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.1.3.2a1.1$x^{3} + 11$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
11.6.1.0a1.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
11.2.3.4a1.2$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 84 x + 19$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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