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Properties

Label 20.0.596...344.3
Degree $20$
Signature $[0, 10]$
Discriminant $5.970\times 10^{27}$
Root discriminant \(24.48\)
Ramified primes $2,11$
Class number $2$
Class group [2]
Galois group $C_2^5:C_{10}$ (as 20T74)

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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 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 8*y^19 + 22*y^18 + 6*y^17 - 147*y^16 + 198*y^15 + 322*y^14 - 948*y^13 + 44*y^12 + 1888*y^11 - 1288*y^10 - 1752*y^9 + 2255*y^8 + 522*y^7 - 1580*y^6 + 286*y^5 + 591*y^4 - 438*y^3 + 132*y^2 - 18*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1)
 

\( x^{20} - 8 x^{19} + 22 x^{18} + 6 x^{17} - 147 x^{16} + 198 x^{15} + 322 x^{14} - 948 x^{13} + 44 x^{12} + \cdots + 1 \) 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:  $20$
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:   \(5969915757478328440239161344\) \(\medspace = 2^{30}\cdot 11^{18}\) 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:  \(24.48\)
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:  $2^{31/16}11^{9/10}\approx 33.15118318510912$
Ramified primes:   \(2\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13\cdots 59}a^{19}+\frac{28\cdots 93}{13\cdots 59}a^{18}+\frac{65\cdots 19}{13\cdots 59}a^{17}-\frac{34\cdots 82}{13\cdots 59}a^{16}-\frac{38\cdots 47}{13\cdots 59}a^{15}-\frac{15\cdots 27}{13\cdots 59}a^{14}+\frac{25\cdots 18}{13\cdots 59}a^{13}-\frac{19\cdots 94}{13\cdots 59}a^{12}-\frac{11\cdots 86}{13\cdots 59}a^{11}+\frac{22\cdots 49}{13\cdots 59}a^{10}+\frac{59\cdots 79}{13\cdots 59}a^{9}-\frac{37\cdots 45}{13\cdots 59}a^{8}+\frac{51\cdots 50}{13\cdots 59}a^{7}+\frac{51\cdots 49}{13\cdots 59}a^{6}+\frac{17\cdots 33}{58\cdots 33}a^{5}+\frac{56\cdots 84}{13\cdots 59}a^{4}-\frac{15\cdots 79}{13\cdots 59}a^{3}+\frac{44\cdots 73}{13\cdots 59}a^{2}+\frac{42\cdots 64}{13\cdots 59}a+\frac{17\cdots 02}{13\cdots 59}$ 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:  $C_{2}$, which has order $2$
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:  $C_{2}$, which has order $2$
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:   $\frac{44\cdots 69}{13\cdots 59}a^{19}-\frac{33\cdots 72}{13\cdots 59}a^{18}+\frac{81\cdots 80}{13\cdots 59}a^{17}+\frac{66\cdots 25}{13\cdots 59}a^{16}-\frac{61\cdots 67}{13\cdots 59}a^{15}+\frac{56\cdots 94}{13\cdots 59}a^{14}+\frac{16\cdots 93}{13\cdots 59}a^{13}-\frac{32\cdots 84}{13\cdots 59}a^{12}-\frac{13\cdots 80}{13\cdots 59}a^{11}+\frac{73\cdots 61}{13\cdots 59}a^{10}-\frac{19\cdots 10}{13\cdots 59}a^{9}-\frac{79\cdots 09}{13\cdots 59}a^{8}+\frac{54\cdots 61}{13\cdots 59}a^{7}+\frac{42\cdots 44}{13\cdots 59}a^{6}-\frac{17\cdots 34}{58\cdots 33}a^{5}-\frac{28\cdots 40}{13\cdots 59}a^{4}+\frac{18\cdots 24}{13\cdots 59}a^{3}-\frac{10\cdots 99}{13\cdots 59}a^{2}+\frac{31\cdots 62}{13\cdots 59}a-\frac{28\cdots 15}{13\cdots 59}$, $\frac{41\cdots 29}{13\cdots 59}a^{19}-\frac{34\cdots 13}{13\cdots 59}a^{18}+\frac{10\cdots 84}{13\cdots 59}a^{17}+\frac{31\cdots 18}{13\cdots 59}a^{16}-\frac{64\cdots 87}{13\cdots 59}a^{15}+\frac{10\cdots 63}{13\cdots 59}a^{14}+\frac{12\cdots 03}{13\cdots 59}a^{13}-\frac{46\cdots 84}{13\cdots 59}a^{12}+\frac{10\cdots 39}{13\cdots 59}a^{11}+\frac{89\cdots 34}{13\cdots 59}a^{10}-\frac{76\cdots 83}{13\cdots 59}a^{9}-\frac{79\cdots 75}{13\cdots 59}a^{8}+\frac{12\cdots 75}{13\cdots 59}a^{7}+\frac{18\cdots 51}{13\cdots 59}a^{6}-\frac{39\cdots 59}{58\cdots 33}a^{5}+\frac{16\cdots 86}{13\cdots 59}a^{4}+\frac{34\cdots 05}{13\cdots 59}a^{3}-\frac{22\cdots 09}{13\cdots 59}a^{2}+\frac{50\cdots 35}{13\cdots 59}a-\frac{39\cdots 63}{13\cdots 59}$, $\frac{35\cdots 45}{13\cdots 59}a^{19}-\frac{27\cdots 04}{13\cdots 59}a^{18}+\frac{73\cdots 72}{13\cdots 59}a^{17}+\frac{32\cdots 70}{13\cdots 59}a^{16}-\frac{51\cdots 79}{13\cdots 59}a^{15}+\frac{62\cdots 39}{13\cdots 59}a^{14}+\frac{12\cdots 78}{13\cdots 59}a^{13}-\frac{31\cdots 32}{13\cdots 59}a^{12}-\frac{41\cdots 06}{13\cdots 59}a^{11}+\frac{67\cdots 46}{13\cdots 59}a^{10}-\frac{33\cdots 15}{13\cdots 59}a^{9}-\frac{70\cdots 76}{13\cdots 59}a^{8}+\frac{67\cdots 38}{13\cdots 59}a^{7}+\frac{33\cdots 68}{13\cdots 59}a^{6}-\frac{21\cdots 20}{58\cdots 33}a^{5}-\frac{22\cdots 87}{13\cdots 59}a^{4}+\frac{19\cdots 46}{13\cdots 59}a^{3}-\frac{11\cdots 17}{13\cdots 59}a^{2}+\frac{30\cdots 21}{13\cdots 59}a-\frac{26\cdots 07}{13\cdots 59}$, $\frac{91\cdots 42}{13\cdots 59}a^{19}-\frac{71\cdots 14}{13\cdots 59}a^{18}+\frac{18\cdots 69}{13\cdots 59}a^{17}+\frac{95\cdots 13}{13\cdots 59}a^{16}-\frac{13\cdots 00}{13\cdots 59}a^{15}+\frac{15\cdots 59}{13\cdots 59}a^{14}+\frac{32\cdots 96}{13\cdots 59}a^{13}-\frac{78\cdots 51}{13\cdots 59}a^{12}-\frac{12\cdots 04}{13\cdots 59}a^{11}+\frac{16\cdots 30}{13\cdots 59}a^{10}-\frac{81\cdots 78}{13\cdots 59}a^{9}-\frac{16\cdots 67}{13\cdots 59}a^{8}+\frac{16\cdots 59}{13\cdots 59}a^{7}+\frac{71\cdots 29}{13\cdots 59}a^{6}-\frac{52\cdots 88}{58\cdots 33}a^{5}+\frac{81\cdots 50}{13\cdots 59}a^{4}+\frac{50\cdots 97}{13\cdots 59}a^{3}-\frac{31\cdots 24}{13\cdots 59}a^{2}+\frac{79\cdots 30}{13\cdots 59}a-\frac{67\cdots 15}{13\cdots 59}$, $\frac{10\cdots 01}{58\cdots 33}a^{19}-\frac{10\cdots 46}{58\cdots 33}a^{18}+\frac{37\cdots 36}{58\cdots 33}a^{17}-\frac{23\cdots 94}{58\cdots 33}a^{16}-\frac{19\cdots 76}{58\cdots 33}a^{15}+\frac{46\cdots 16}{58\cdots 33}a^{14}+\frac{18\cdots 16}{58\cdots 33}a^{13}-\frac{18\cdots 96}{58\cdots 33}a^{12}+\frac{12\cdots 33}{58\cdots 33}a^{11}+\frac{30\cdots 10}{58\cdots 33}a^{10}-\frac{43\cdots 86}{58\cdots 33}a^{9}-\frac{20\cdots 64}{58\cdots 33}a^{8}+\frac{61\cdots 48}{58\cdots 33}a^{7}-\frac{61\cdots 90}{58\cdots 33}a^{6}-\frac{44\cdots 98}{58\cdots 33}a^{5}+\frac{13\cdots 32}{58\cdots 33}a^{4}+\frac{14\cdots 37}{58\cdots 33}a^{3}-\frac{10\cdots 40}{58\cdots 33}a^{2}+\frac{28\cdots 06}{58\cdots 33}a-\frac{34\cdots 98}{58\cdots 33}$, $\frac{79\cdots 18}{13\cdots 59}a^{19}-\frac{57\cdots 70}{13\cdots 59}a^{18}+\frac{13\cdots 42}{13\cdots 59}a^{17}+\frac{15\cdots 23}{13\cdots 59}a^{16}-\frac{10\cdots 41}{13\cdots 59}a^{15}+\frac{74\cdots 37}{13\cdots 59}a^{14}+\frac{33\cdots 16}{13\cdots 59}a^{13}-\frac{52\cdots 44}{13\cdots 59}a^{12}-\frac{41\cdots 04}{13\cdots 59}a^{11}+\frac{13\cdots 19}{13\cdots 59}a^{10}+\frac{17\cdots 81}{13\cdots 59}a^{9}-\frac{16\cdots 87}{13\cdots 59}a^{8}+\frac{58\cdots 51}{13\cdots 59}a^{7}+\frac{11\cdots 31}{13\cdots 59}a^{6}-\frac{23\cdots 75}{58\cdots 33}a^{5}-\frac{40\cdots 60}{13\cdots 59}a^{4}+\frac{30\cdots 93}{13\cdots 59}a^{3}-\frac{50\cdots 02}{13\cdots 59}a^{2}-\frac{26\cdots 18}{13\cdots 59}a+\frac{13\cdots 22}{13\cdots 59}$, $\frac{70\cdots 25}{13\cdots 59}a^{19}-\frac{55\cdots 07}{13\cdots 59}a^{18}+\frac{14\cdots 86}{13\cdots 59}a^{17}+\frac{70\cdots 44}{13\cdots 59}a^{16}-\frac{10\cdots 36}{13\cdots 59}a^{15}+\frac{11\cdots 27}{13\cdots 59}a^{14}+\frac{25\cdots 13}{13\cdots 59}a^{13}-\frac{62\cdots 15}{13\cdots 59}a^{12}-\frac{89\cdots 55}{13\cdots 59}a^{11}+\frac{13\cdots 63}{13\cdots 59}a^{10}-\frac{65\cdots 75}{13\cdots 59}a^{9}-\frac{13\cdots 06}{13\cdots 59}a^{8}+\frac{13\cdots 55}{13\cdots 59}a^{7}+\frac{62\cdots 00}{13\cdots 59}a^{6}-\frac{42\cdots 88}{58\cdots 33}a^{5}+\frac{68\cdots 85}{13\cdots 59}a^{4}+\frac{41\cdots 45}{13\cdots 59}a^{3}-\frac{22\cdots 16}{13\cdots 59}a^{2}+\frac{44\cdots 18}{13\cdots 59}a-\frac{24\cdots 77}{13\cdots 59}$, $\frac{76\cdots 72}{13\cdots 59}a^{19}-\frac{60\cdots 84}{13\cdots 59}a^{18}+\frac{16\cdots 37}{13\cdots 59}a^{17}+\frac{64\cdots 42}{13\cdots 59}a^{16}-\frac{11\cdots 39}{13\cdots 59}a^{15}+\frac{13\cdots 93}{13\cdots 59}a^{14}+\frac{26\cdots 37}{13\cdots 59}a^{13}-\frac{69\cdots 12}{13\cdots 59}a^{12}-\frac{46\cdots 36}{13\cdots 59}a^{11}+\frac{14\cdots 41}{13\cdots 59}a^{10}-\frac{81\cdots 60}{13\cdots 59}a^{9}-\frac{14\cdots 82}{13\cdots 59}a^{8}+\frac{15\cdots 36}{13\cdots 59}a^{7}+\frac{57\cdots 05}{13\cdots 59}a^{6}-\frac{48\cdots 19}{58\cdots 33}a^{5}+\frac{92\cdots 16}{13\cdots 59}a^{4}+\frac{44\cdots 93}{13\cdots 59}a^{3}-\frac{28\cdots 31}{13\cdots 59}a^{2}+\frac{73\cdots 17}{13\cdots 59}a-\frac{74\cdots 73}{13\cdots 59}$, $\frac{10\cdots 67}{13\cdots 59}a^{19}-\frac{84\cdots 52}{13\cdots 59}a^{18}+\frac{23\cdots 49}{13\cdots 59}a^{17}+\frac{71\cdots 16}{13\cdots 59}a^{16}-\frac{15\cdots 41}{13\cdots 59}a^{15}+\frac{20\cdots 15}{13\cdots 59}a^{14}+\frac{35\cdots 34}{13\cdots 59}a^{13}-\frac{99\cdots 45}{13\cdots 59}a^{12}+\frac{12\cdots 70}{13\cdots 59}a^{11}+\frac{20\cdots 50}{13\cdots 59}a^{10}-\frac{13\cdots 51}{13\cdots 59}a^{9}-\frac{19\cdots 37}{13\cdots 59}a^{8}+\frac{23\cdots 90}{13\cdots 59}a^{7}+\frac{64\cdots 51}{13\cdots 59}a^{6}-\frac{72\cdots 60}{58\cdots 33}a^{5}+\frac{24\cdots 81}{13\cdots 59}a^{4}+\frac{64\cdots 81}{13\cdots 59}a^{3}-\frac{44\cdots 40}{13\cdots 59}a^{2}+\frac{12\cdots 92}{13\cdots 59}a-\frac{12\cdots 60}{13\cdots 59}$ Copy content Toggle raw display
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:  \( 159636.716542 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 159636.716542 \cdot 2}{2\cdot\sqrt{5969915757478328440239161344}}\cr\approx \mathstrut & 0.198128690295 \end{aligned}\]

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 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*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^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*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^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*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 = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^19 + 22*x^18 + 6*x^17 - 147*x^16 + 198*x^15 + 322*x^14 - 948*x^13 + 44*x^12 + 1888*x^11 - 1288*x^10 - 1752*x^9 + 2255*x^8 + 522*x^7 - 1580*x^6 + 286*x^5 + 591*x^4 - 438*x^3 + 132*x^2 - 18*x + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^5:C_{10}$ (as 20T74):

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 320
The 32 conjugacy class representatives for $C_2^5:C_{10}$
Character table for $C_2^5:C_{10}$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.2414538435584.1, 10.2.2414538435584.1, 10.4.219503494144.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 20.8.5969915757478328440239161344.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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 2.5.4.30a10.2$x^{20} + 6 x^{17} + 6 x^{15} + 12 x^{14} + 24 x^{12} + 10 x^{11} + 14 x^{10} + 30 x^{9} + 3 x^{8} + 34 x^{7} + 12 x^{6} + 14 x^{5} + 20 x^{4} + 16 x^{2} + 4 x + 7$$4$$5$$30$20T74not computed
\(11\) Copy content Toggle raw display 11.2.10.18a1.2$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35840 x + 1035$$10$$2$$18$20T3$$[\ ]_{10}^{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)