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Properties

Label 20.0.219...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $2.191\times 10^{40}$
Root discriminant \(104.00\)
Ramified primes $2,3,5,11$
Class number $4$ (GRH)
Class group [4] (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 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072)
 
Copy content gp:K = bnfinit(y^20 - 40*y^18 - 80*y^17 + 540*y^16 + 1920*y^15 - 1120*y^14 - 12280*y^13 + 54950*y^12 + 91520*y^11 - 398016*y^10 + 528960*y^9 + 1490240*y^8 - 4294560*y^7 + 2844720*y^6 + 12388192*y^5 + 10739860*y^4 - 24257280*y^3 + 11562000*y^2 + 174841920*y + 234225072, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072)
 

\( x^{20} - 40 x^{18} - 80 x^{17} + 540 x^{16} + 1920 x^{15} - 1120 x^{14} - 12280 x^{13} + \cdots + 234225072 \) 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:   \(21913334086662881280000000000000000000000\) \(\medspace = 2^{59}\cdot 3^{2}\cdot 5^{22}\cdot 11^{6}\) 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:  \(104.00\)
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:   \(2\), \(3\), \(5\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{4}a^{16}$, $\frac{1}{4}a^{17}$, $\frac{1}{408}a^{18}+\frac{2}{17}a^{17}-\frac{23}{204}a^{16}+\frac{13}{102}a^{15}-\frac{2}{17}a^{14}+\frac{5}{34}a^{13}+\frac{4}{51}a^{12}-\frac{2}{51}a^{11}+\frac{25}{204}a^{10}-\frac{11}{51}a^{9}+\frac{1}{34}a^{8}+\frac{2}{17}a^{7}+\frac{13}{51}a^{6}-\frac{7}{17}a^{5}-\frac{4}{17}a^{4}+\frac{5}{51}a^{3}+\frac{43}{102}a^{2}-\frac{6}{17}a+\frac{8}{17}$, $\frac{1}{13\cdots 04}a^{19}+\frac{49\cdots 85}{67\cdots 52}a^{18}-\frac{34\cdots 43}{67\cdots 52}a^{17}-\frac{50\cdots 81}{49\cdots 57}a^{16}+\frac{82\cdots 61}{16\cdots 38}a^{15}+\frac{57\cdots 43}{56\cdots 46}a^{14}+\frac{96\cdots 64}{84\cdots 69}a^{13}-\frac{25\cdots 61}{56\cdots 46}a^{12}+\frac{54\cdots 29}{22\cdots 84}a^{11}+\frac{21\cdots 71}{11\cdots 92}a^{10}+\frac{77\cdots 75}{33\cdots 76}a^{9}-\frac{12\cdots 69}{56\cdots 46}a^{8}-\frac{99\cdots 11}{16\cdots 38}a^{7}+\frac{17\cdots 91}{84\cdots 69}a^{6}+\frac{90\cdots 39}{28\cdots 23}a^{5}-\frac{32\cdots 21}{84\cdots 69}a^{4}+\frac{53\cdots 03}{11\cdots 92}a^{3}+\frac{79\cdots 39}{16\cdots 38}a^{2}+\frac{12\cdots 25}{56\cdots 46}a+\frac{56\cdots 76}{28\cdots 23}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (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:  $C_{4}$, which has order $4$ (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:   $\frac{44\cdots 99}{58\cdots 89}a^{19}-\frac{47\cdots 53}{11\cdots 78}a^{18}-\frac{25\cdots 11}{58\cdots 89}a^{17}+\frac{20\cdots 83}{23\cdots 56}a^{16}+\frac{76\cdots 28}{58\cdots 89}a^{15}+\frac{66\cdots 22}{58\cdots 89}a^{14}-\frac{93\cdots 57}{58\cdots 89}a^{13}-\frac{60\cdots 97}{11\cdots 78}a^{12}+\frac{35\cdots 58}{58\cdots 89}a^{11}+\frac{11\cdots 21}{58\cdots 89}a^{10}-\frac{21\cdots 26}{58\cdots 89}a^{9}+\frac{41\cdots 67}{58\cdots 89}a^{8}+\frac{15\cdots 00}{58\cdots 89}a^{7}-\frac{55\cdots 46}{58\cdots 89}a^{6}+\frac{16\cdots 36}{58\cdots 89}a^{5}-\frac{19\cdots 86}{58\cdots 89}a^{4}+\frac{19\cdots 64}{58\cdots 89}a^{3}-\frac{75\cdots 16}{58\cdots 89}a^{2}-\frac{19\cdots 84}{58\cdots 89}a-\frac{22\cdots 31}{58\cdots 89}$, $\frac{28\cdots 22}{58\cdots 89}a^{19}-\frac{12\cdots 28}{58\cdots 89}a^{18}-\frac{10\cdots 50}{58\cdots 89}a^{17}+\frac{23\cdots 80}{58\cdots 89}a^{16}+\frac{20\cdots 92}{58\cdots 89}a^{15}-\frac{10\cdots 59}{58\cdots 89}a^{14}-\frac{18\cdots 35}{58\cdots 89}a^{13}-\frac{24\cdots 39}{11\cdots 78}a^{12}+\frac{24\cdots 94}{58\cdots 89}a^{11}-\frac{53\cdots 01}{58\cdots 89}a^{10}-\frac{13\cdots 78}{58\cdots 89}a^{9}+\frac{57\cdots 05}{58\cdots 89}a^{8}-\frac{94\cdots 04}{58\cdots 89}a^{7}-\frac{92\cdots 76}{58\cdots 89}a^{6}+\frac{66\cdots 50}{58\cdots 89}a^{5}-\frac{92\cdots 31}{58\cdots 89}a^{4}+\frac{24\cdots 40}{58\cdots 89}a^{3}-\frac{95\cdots 24}{58\cdots 89}a^{2}+\frac{12\cdots 16}{58\cdots 89}a-\frac{85\cdots 93}{58\cdots 89}$, $\frac{41\cdots 41}{28\cdots 23}a^{19}-\frac{21\cdots 55}{67\cdots 52}a^{18}-\frac{18\cdots 93}{11\cdots 92}a^{17}+\frac{19\cdots 27}{19\cdots 28}a^{16}+\frac{12\cdots 73}{16\cdots 38}a^{15}+\frac{11\cdots 79}{56\cdots 46}a^{14}-\frac{27\cdots 14}{28\cdots 23}a^{13}-\frac{50\cdots 49}{16\cdots 38}a^{12}+\frac{84\cdots 07}{16\cdots 38}a^{11}-\frac{10\cdots 75}{33\cdots 76}a^{10}-\frac{83\cdots 41}{16\cdots 38}a^{9}+\frac{45\cdots 68}{28\cdots 23}a^{8}+\frac{28\cdots 86}{28\cdots 23}a^{7}-\frac{31\cdots 13}{84\cdots 69}a^{6}-\frac{78\cdots 40}{28\cdots 23}a^{5}+\frac{24\cdots 65}{28\cdots 23}a^{4}-\frac{11\cdots 25}{84\cdots 69}a^{3}-\frac{21\cdots 11}{16\cdots 38}a^{2}-\frac{68\cdots 57}{28\cdots 23}a-\frac{50\cdots 96}{28\cdots 23}$, $\frac{36\cdots 73}{67\cdots 52}a^{19}-\frac{68\cdots 41}{67\cdots 52}a^{18}-\frac{69\cdots 93}{33\cdots 76}a^{17}-\frac{57\cdots 73}{19\cdots 28}a^{16}+\frac{53\cdots 71}{16\cdots 38}a^{15}+\frac{12\cdots 14}{28\cdots 23}a^{14}-\frac{13\cdots 58}{84\cdots 69}a^{13}-\frac{10\cdots 62}{28\cdots 23}a^{12}+\frac{41\cdots 29}{11\cdots 92}a^{11}-\frac{20\cdots 23}{11\cdots 92}a^{10}-\frac{38\cdots 65}{16\cdots 38}a^{9}+\frac{23\cdots 45}{28\cdots 23}a^{8}-\frac{88\cdots 41}{84\cdots 69}a^{7}+\frac{26\cdots 32}{84\cdots 69}a^{6}+\frac{31\cdots 44}{28\cdots 23}a^{5}+\frac{36\cdots 94}{84\cdots 69}a^{4}-\frac{17\cdots 37}{56\cdots 46}a^{3}-\frac{10\cdots 21}{16\cdots 38}a^{2}+\frac{42\cdots 47}{28\cdots 23}a+\frac{15\cdots 02}{28\cdots 23}$, $\frac{14\cdots 61}{56\cdots 46}a^{19}-\frac{81\cdots 05}{16\cdots 38}a^{18}-\frac{26\cdots 42}{28\cdots 23}a^{17}-\frac{35\cdots 27}{19\cdots 28}a^{16}+\frac{24\cdots 51}{16\cdots 38}a^{15}+\frac{56\cdots 75}{28\cdots 23}a^{14}-\frac{42\cdots 31}{56\cdots 46}a^{13}-\frac{14\cdots 11}{84\cdots 69}a^{12}+\frac{14\cdots 91}{84\cdots 69}a^{11}-\frac{17\cdots 13}{16\cdots 38}a^{10}-\frac{77\cdots 78}{84\cdots 69}a^{9}+\frac{91\cdots 69}{28\cdots 23}a^{8}-\frac{54\cdots 10}{28\cdots 23}a^{7}-\frac{80\cdots 15}{84\cdots 69}a^{6}+\frac{79\cdots 82}{28\cdots 23}a^{5}-\frac{51\cdots 70}{28\cdots 23}a^{4}+\frac{32\cdots 18}{84\cdots 69}a^{3}-\frac{87\cdots 78}{84\cdots 69}a^{2}+\frac{50\cdots 68}{28\cdots 23}a+\frac{57\cdots 91}{28\cdots 23}$, $\frac{17\cdots 05}{84\cdots 69}a^{19}-\frac{21\cdots 88}{84\cdots 69}a^{18}-\frac{16\cdots 52}{84\cdots 69}a^{17}+\frac{46\cdots 22}{97\cdots 07}a^{16}+\frac{57\cdots 09}{84\cdots 69}a^{15}+\frac{34\cdots 22}{28\cdots 23}a^{14}-\frac{41\cdots 00}{84\cdots 69}a^{13}-\frac{44\cdots 59}{16\cdots 38}a^{12}-\frac{28\cdots 94}{84\cdots 69}a^{11}-\frac{30\cdots 37}{16\cdots 38}a^{10}-\frac{80\cdots 21}{84\cdots 69}a^{9}-\frac{99\cdots 25}{56\cdots 46}a^{8}-\frac{34\cdots 74}{84\cdots 69}a^{7}-\frac{11\cdots 19}{84\cdots 69}a^{6}-\frac{66\cdots 12}{28\cdots 23}a^{5}-\frac{30\cdots 76}{84\cdots 69}a^{4}-\frac{60\cdots 36}{84\cdots 69}a^{3}-\frac{13\cdots 08}{84\cdots 69}a^{2}-\frac{63\cdots 12}{28\cdots 23}a-\frac{42\cdots 43}{28\cdots 23}$, $\frac{33\cdots 55}{67\cdots 52}a^{19}-\frac{83\cdots 11}{16\cdots 38}a^{18}-\frac{31\cdots 87}{16\cdots 38}a^{17}-\frac{20\cdots 31}{99\cdots 14}a^{16}+\frac{44\cdots 89}{16\cdots 38}a^{15}+\frac{35\cdots 15}{56\cdots 46}a^{14}-\frac{70\cdots 40}{84\cdots 69}a^{13}-\frac{97\cdots 80}{28\cdots 23}a^{12}+\frac{34\cdots 69}{11\cdots 92}a^{11}-\frac{13\cdots 47}{56\cdots 46}a^{10}-\frac{17\cdots 34}{84\cdots 69}a^{9}+\frac{28\cdots 47}{56\cdots 46}a^{8}+\frac{33\cdots 25}{84\cdots 69}a^{7}-\frac{20\cdots 96}{84\cdots 69}a^{6}+\frac{10\cdots 18}{28\cdots 23}a^{5}+\frac{36\cdots 29}{84\cdots 69}a^{4}-\frac{56\cdots 81}{56\cdots 46}a^{3}+\frac{17\cdots 04}{84\cdots 69}a^{2}+\frac{74\cdots 80}{28\cdots 23}a+\frac{24\cdots 03}{28\cdots 23}$, $\frac{98\cdots 03}{22\cdots 84}a^{19}-\frac{77\cdots 07}{11\cdots 92}a^{18}-\frac{53\cdots 07}{28\cdots 23}a^{17}-\frac{21\cdots 95}{16\cdots 19}a^{16}+\frac{98\cdots 69}{28\cdots 23}a^{15}+\frac{22\cdots 02}{28\cdots 23}a^{14}-\frac{11\cdots 67}{56\cdots 46}a^{13}-\frac{29\cdots 30}{28\cdots 23}a^{12}+\frac{19\cdots 51}{11\cdots 92}a^{11}+\frac{41\cdots 89}{28\cdots 23}a^{10}-\frac{50\cdots 00}{28\cdots 23}a^{9}+\frac{24\cdots 21}{56\cdots 46}a^{8}+\frac{93\cdots 51}{28\cdots 23}a^{7}-\frac{80\cdots 07}{28\cdots 23}a^{6}+\frac{43\cdots 22}{28\cdots 23}a^{5}+\frac{89\cdots 39}{28\cdots 23}a^{4}+\frac{40\cdots 65}{56\cdots 46}a^{3}-\frac{56\cdots 91}{28\cdots 23}a^{2}-\frac{90\cdots 44}{28\cdots 23}a+\frac{15\cdots 57}{28\cdots 23}$, $\frac{31\cdots 09}{33\cdots 76}a^{19}+\frac{13\cdots 57}{11\cdots 92}a^{18}-\frac{70\cdots 65}{16\cdots 38}a^{17}-\frac{15\cdots 51}{29\cdots 21}a^{16}-\frac{51\cdots 70}{28\cdots 23}a^{15}+\frac{10\cdots 03}{28\cdots 23}a^{14}+\frac{31\cdots 53}{84\cdots 69}a^{13}+\frac{62\cdots 83}{84\cdots 69}a^{12}-\frac{10\cdots 01}{16\cdots 38}a^{11}+\frac{41\cdots 15}{16\cdots 38}a^{10}+\frac{36\cdots 75}{28\cdots 23}a^{9}-\frac{13\cdots 80}{28\cdots 23}a^{8}-\frac{78\cdots 12}{84\cdots 69}a^{7}+\frac{26\cdots 68}{28\cdots 23}a^{6}-\frac{13\cdots 62}{28\cdots 23}a^{5}-\frac{16\cdots 20}{84\cdots 69}a^{4}+\frac{75\cdots 89}{84\cdots 69}a^{3}+\frac{12\cdots 11}{28\cdots 23}a^{2}+\frac{20\cdots 06}{28\cdots 23}a+\frac{13\cdots 81}{28\cdots 23}$ 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:  \( 1013484687050 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1013484687050 \cdot 4}{2\cdot\sqrt{21913334086662881280000000000000000000000}}\cr\approx \mathstrut & 1.31308137270483 \end{aligned}\] (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 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072) 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 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072, 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 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072); 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 - 40*x^18 - 80*x^17 + 540*x^16 + 1920*x^15 - 1120*x^14 - 12280*x^13 + 54950*x^12 + 91520*x^11 - 398016*x^10 + 528960*x^9 + 1490240*x^8 - 4294560*x^7 + 2844720*x^6 + 12388192*x^5 + 10739860*x^4 - 24257280*x^3 + 11562000*x^2 + 174841920*x + 234225072); 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 R R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $20$ $15{,}\,{\href{/padicField/31.5.0.1}{5} }$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ $20$

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 2.1.4.11a1.9$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$$[3, 4]$$
2.2.8.48b7.54$x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 406 x^{12} + 920 x^{11} + 1716 x^{10} + 2664 x^{9} + 3483 x^{8} + 3848 x^{7} + 3612 x^{6} + 2864 x^{5} + 1914 x^{4} + 1056 x^{3} + 468 x^{2} + 160 x + 31$$8$$2$$48$16T1385not computed
\(3\) Copy content Toggle raw display 3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.12.1.0a1.1$x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$
\(5\) 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$$$[\ ]_{3}^{2}$$
11.2.3.4a1.2$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 84 x + 19$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
11.6.1.0a1.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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