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Properties

Label 20.2.141...408.1
Degree $20$
Signature $(2, 9)$
Discriminant $-1.418\times 10^{27}$
Root discriminant \(22.78\)
Ramified primes $2,3,43$
Class number $3$
Class group [3]
Galois group $S_5$ (as 20T35)

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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 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*x - 1)
 
Copy content gp:K = bnfinit(y^20 - 7*y^19 + 19*y^18 - 5*y^17 - 108*y^16 + 311*y^15 - 333*y^14 - 167*y^13 + 992*y^12 - 1203*y^11 + 141*y^10 + 1555*y^9 - 2554*y^8 + 2293*y^7 - 1387*y^6 + 609*y^5 - 208*y^4 + 77*y^3 - 33*y^2 + 9*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*x - 1)
 

\( x^{20} - 7 x^{19} + 19 x^{18} - 5 x^{17} - 108 x^{16} + 311 x^{15} - 333 x^{14} - 167 x^{13} + 992 x^{12} + \cdots - 1 \) 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:  $(2, 9)$
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:   \(-1417868996949216476646801408\) \(\medspace = -\,2^{16}\cdot 3^{16}\cdot 43^{9}\) 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:  \(22.78\)
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:  $2^{4/5}3^{4/5}43^{1/2}\approx 27.49509522275834$
Ramified primes:   \(2\), \(3\), \(43\) 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:  \(\Q(\sqrt{-43}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{50\cdots 19}a^{19}-\frac{17\cdots 54}{50\cdots 19}a^{18}-\frac{12\cdots 81}{50\cdots 19}a^{17}+\frac{15\cdots 03}{50\cdots 19}a^{16}-\frac{91\cdots 22}{50\cdots 19}a^{15}-\frac{23\cdots 75}{50\cdots 19}a^{14}+\frac{14\cdots 68}{50\cdots 19}a^{13}+\frac{12\cdots 09}{50\cdots 19}a^{12}-\frac{20\cdots 10}{50\cdots 19}a^{11}+\frac{16\cdots 28}{50\cdots 19}a^{10}+\frac{89\cdots 00}{50\cdots 19}a^{9}+\frac{67\cdots 09}{50\cdots 19}a^{8}-\frac{73\cdots 81}{50\cdots 19}a^{7}-\frac{98\cdots 38}{50\cdots 19}a^{6}+\frac{19\cdots 96}{50\cdots 19}a^{5}+\frac{11\cdots 75}{50\cdots 19}a^{4}+\frac{11\cdots 86}{50\cdots 19}a^{3}+\frac{13\cdots 17}{50\cdots 19}a^{2}-\frac{11\cdots 47}{50\cdots 19}a+\frac{11\cdots 90}{50\cdots 19}$ 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:  $C_{3}$, which has order $3$
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:  $C_{3}$, which has order $3$
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:  $10$
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:   $\frac{38\cdots 33}{50\cdots 19}a^{19}-\frac{27\cdots 35}{50\cdots 19}a^{18}+\frac{78\cdots 24}{50\cdots 19}a^{17}-\frac{30\cdots 31}{50\cdots 19}a^{16}-\frac{43\cdots 34}{50\cdots 19}a^{15}+\frac{13\cdots 83}{50\cdots 19}a^{14}-\frac{14\cdots 58}{50\cdots 19}a^{13}-\frac{65\cdots 29}{50\cdots 19}a^{12}+\frac{42\cdots 44}{50\cdots 19}a^{11}-\frac{52\cdots 73}{50\cdots 19}a^{10}+\frac{56\cdots 27}{50\cdots 19}a^{9}+\frac{68\cdots 67}{50\cdots 19}a^{8}-\frac{10\cdots 62}{50\cdots 19}a^{7}+\frac{93\cdots 89}{50\cdots 19}a^{6}-\frac{51\cdots 68}{50\cdots 19}a^{5}+\frac{20\cdots 86}{50\cdots 19}a^{4}-\frac{62\cdots 57}{50\cdots 19}a^{3}+\frac{24\cdots 22}{50\cdots 19}a^{2}-\frac{12\cdots 21}{50\cdots 19}a+\frac{25\cdots 85}{50\cdots 19}$, $\frac{63\cdots 59}{50\cdots 19}a^{19}-\frac{40\cdots 80}{50\cdots 19}a^{18}+\frac{96\cdots 47}{50\cdots 19}a^{17}+\frac{26\cdots 53}{50\cdots 19}a^{16}-\frac{67\cdots 94}{50\cdots 19}a^{15}+\frac{15\cdots 53}{50\cdots 19}a^{14}-\frac{11\cdots 00}{50\cdots 19}a^{13}-\frac{18\cdots 36}{50\cdots 19}a^{12}+\frac{53\cdots 56}{50\cdots 19}a^{11}-\frac{44\cdots 49}{50\cdots 19}a^{10}-\frac{20\cdots 20}{50\cdots 19}a^{9}+\frac{90\cdots 11}{50\cdots 19}a^{8}-\frac{10\cdots 06}{50\cdots 19}a^{7}+\frac{75\cdots 05}{50\cdots 19}a^{6}-\frac{34\cdots 97}{50\cdots 19}a^{5}+\frac{11\cdots 85}{50\cdots 19}a^{4}-\frac{30\cdots 53}{50\cdots 19}a^{3}+\frac{18\cdots 49}{50\cdots 19}a^{2}-\frac{71\cdots 23}{50\cdots 19}a+\frac{51\cdots 37}{50\cdots 19}$, $\frac{48\cdots 98}{50\cdots 19}a^{19}-\frac{32\cdots 13}{50\cdots 19}a^{18}+\frac{80\cdots 55}{50\cdots 19}a^{17}+\frac{65\cdots 29}{50\cdots 19}a^{16}-\frac{52\cdots 85}{50\cdots 19}a^{15}+\frac{13\cdots 66}{50\cdots 19}a^{14}-\frac{11\cdots 27}{50\cdots 19}a^{13}-\frac{12\cdots 92}{50\cdots 19}a^{12}+\frac{43\cdots 16}{50\cdots 19}a^{11}-\frac{41\cdots 65}{50\cdots 19}a^{10}-\frac{89\cdots 10}{50\cdots 19}a^{9}+\frac{71\cdots 64}{50\cdots 19}a^{8}-\frac{96\cdots 63}{50\cdots 19}a^{7}+\frac{75\cdots 30}{50\cdots 19}a^{6}-\frac{41\cdots 19}{50\cdots 19}a^{5}+\frac{16\cdots 88}{50\cdots 19}a^{4}-\frac{55\cdots 10}{50\cdots 19}a^{3}+\frac{22\cdots 22}{50\cdots 19}a^{2}-\frac{77\cdots 33}{50\cdots 19}a+\frac{85\cdots 78}{50\cdots 19}$, $a$, $\frac{24\cdots 26}{50\cdots 19}a^{19}-\frac{12\cdots 45}{50\cdots 19}a^{18}+\frac{17\cdots 23}{50\cdots 19}a^{17}+\frac{57\cdots 84}{50\cdots 19}a^{16}-\frac{24\cdots 60}{50\cdots 19}a^{15}+\frac{26\cdots 70}{50\cdots 19}a^{14}+\frac{30\cdots 58}{50\cdots 19}a^{13}-\frac{11\cdots 07}{50\cdots 19}a^{12}+\frac{10\cdots 12}{50\cdots 19}a^{11}+\frac{83\cdots 24}{50\cdots 19}a^{10}-\frac{26\cdots 47}{50\cdots 19}a^{9}+\frac{21\cdots 44}{50\cdots 19}a^{8}+\frac{13\cdots 56}{50\cdots 19}a^{7}-\frac{18\cdots 84}{50\cdots 19}a^{6}+\frac{17\cdots 71}{50\cdots 19}a^{5}-\frac{92\cdots 01}{50\cdots 19}a^{4}+\frac{32\cdots 04}{50\cdots 19}a^{3}-\frac{54\cdots 73}{50\cdots 19}a^{2}+\frac{56\cdots 98}{50\cdots 19}a-\frac{15\cdots 29}{50\cdots 19}$, $\frac{49\cdots 37}{50\cdots 19}a^{19}-\frac{34\cdots 32}{50\cdots 19}a^{18}+\frac{90\cdots 90}{50\cdots 19}a^{17}-\frac{12\cdots 61}{50\cdots 19}a^{16}-\frac{54\cdots 54}{50\cdots 19}a^{15}+\frac{14\cdots 52}{50\cdots 19}a^{14}-\frac{14\cdots 40}{50\cdots 19}a^{13}-\frac{11\cdots 37}{50\cdots 19}a^{12}+\frac{49\cdots 18}{50\cdots 19}a^{11}-\frac{52\cdots 47}{50\cdots 19}a^{10}-\frac{38\cdots 25}{50\cdots 19}a^{9}+\frac{81\cdots 47}{50\cdots 19}a^{8}-\frac{11\cdots 82}{50\cdots 19}a^{7}+\frac{92\cdots 86}{50\cdots 19}a^{6}-\frac{49\cdots 57}{50\cdots 19}a^{5}+\frac{18\cdots 79}{50\cdots 19}a^{4}-\frac{57\cdots 38}{50\cdots 19}a^{3}+\frac{24\cdots 54}{50\cdots 19}a^{2}-\frac{11\cdots 09}{50\cdots 19}a+\frac{21\cdots 52}{50\cdots 19}$, $\frac{24\cdots 26}{50\cdots 19}a^{19}-\frac{12\cdots 45}{50\cdots 19}a^{18}+\frac{17\cdots 23}{50\cdots 19}a^{17}+\frac{57\cdots 84}{50\cdots 19}a^{16}-\frac{24\cdots 60}{50\cdots 19}a^{15}+\frac{26\cdots 70}{50\cdots 19}a^{14}+\frac{30\cdots 58}{50\cdots 19}a^{13}-\frac{11\cdots 07}{50\cdots 19}a^{12}+\frac{10\cdots 12}{50\cdots 19}a^{11}+\frac{83\cdots 24}{50\cdots 19}a^{10}-\frac{26\cdots 47}{50\cdots 19}a^{9}+\frac{21\cdots 44}{50\cdots 19}a^{8}+\frac{13\cdots 56}{50\cdots 19}a^{7}-\frac{18\cdots 84}{50\cdots 19}a^{6}+\frac{17\cdots 71}{50\cdots 19}a^{5}-\frac{92\cdots 01}{50\cdots 19}a^{4}+\frac{32\cdots 04}{50\cdots 19}a^{3}-\frac{54\cdots 73}{50\cdots 19}a^{2}+\frac{61\cdots 17}{50\cdots 19}a-\frac{20\cdots 48}{50\cdots 19}$, $\frac{10\cdots 62}{50\cdots 19}a^{19}-\frac{69\cdots 87}{50\cdots 19}a^{18}+\frac{17\cdots 09}{50\cdots 19}a^{17}+\frac{15\cdots 05}{50\cdots 19}a^{16}-\frac{11\cdots 29}{50\cdots 19}a^{15}+\frac{28\cdots 03}{50\cdots 19}a^{14}-\frac{24\cdots 59}{50\cdots 19}a^{13}-\frac{27\cdots 22}{50\cdots 19}a^{12}+\frac{94\cdots 39}{50\cdots 19}a^{11}-\frac{90\cdots 43}{50\cdots 19}a^{10}-\frac{21\cdots 90}{50\cdots 19}a^{9}+\frac{15\cdots 69}{50\cdots 19}a^{8}-\frac{20\cdots 57}{50\cdots 19}a^{7}+\frac{15\cdots 96}{50\cdots 19}a^{6}-\frac{81\cdots 71}{50\cdots 19}a^{5}+\frac{30\cdots 11}{50\cdots 19}a^{4}-\frac{89\cdots 62}{50\cdots 19}a^{3}+\frac{41\cdots 93}{50\cdots 19}a^{2}-\frac{17\cdots 46}{50\cdots 19}a+\frac{20\cdots 15}{50\cdots 19}$, $\frac{36\cdots 55}{50\cdots 19}a^{19}-\frac{25\cdots 17}{50\cdots 19}a^{18}+\frac{66\cdots 48}{50\cdots 19}a^{17}-\frac{95\cdots 57}{50\cdots 19}a^{16}-\frac{39\cdots 00}{50\cdots 19}a^{15}+\frac{10\cdots 90}{50\cdots 19}a^{14}-\frac{10\cdots 96}{50\cdots 19}a^{13}-\frac{76\cdots 43}{50\cdots 19}a^{12}+\frac{35\cdots 75}{50\cdots 19}a^{11}-\frac{39\cdots 88}{50\cdots 19}a^{10}-\frac{49\cdots 10}{50\cdots 19}a^{9}+\frac{57\cdots 97}{50\cdots 19}a^{8}-\frac{85\cdots 05}{50\cdots 19}a^{7}+\frac{72\cdots 41}{50\cdots 19}a^{6}-\frac{40\cdots 40}{50\cdots 19}a^{5}+\frac{16\cdots 59}{50\cdots 19}a^{4}-\frac{49\cdots 72}{50\cdots 19}a^{3}+\frac{18\cdots 79}{50\cdots 19}a^{2}-\frac{88\cdots 31}{50\cdots 19}a+\frac{14\cdots 37}{50\cdots 19}$, $\frac{82\cdots 85}{50\cdots 19}a^{19}-\frac{53\cdots 74}{50\cdots 19}a^{18}+\frac{12\cdots 14}{50\cdots 19}a^{17}+\frac{28\cdots 30}{50\cdots 19}a^{16}-\frac{87\cdots 02}{50\cdots 19}a^{15}+\frac{20\cdots 68}{50\cdots 19}a^{14}-\frac{16\cdots 26}{50\cdots 19}a^{13}-\frac{22\cdots 70}{50\cdots 19}a^{12}+\frac{69\cdots 53}{50\cdots 19}a^{11}-\frac{61\cdots 35}{50\cdots 19}a^{10}-\frac{21\cdots 74}{50\cdots 19}a^{9}+\frac{11\cdots 23}{50\cdots 19}a^{8}-\frac{14\cdots 60}{50\cdots 19}a^{7}+\frac{10\cdots 13}{50\cdots 19}a^{6}-\frac{55\cdots 70}{50\cdots 19}a^{5}+\frac{20\cdots 28}{50\cdots 19}a^{4}-\frac{63\cdots 13}{50\cdots 19}a^{3}+\frac{29\cdots 00}{50\cdots 19}a^{2}-\frac{10\cdots 29}{50\cdots 19}a+\frac{96\cdots 44}{50\cdots 19}$ 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:  \( 121399.279846 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 121399.279846 \cdot 3}{2\cdot\sqrt{1417868996949216476646801408}}\cr\approx \mathstrut & 0.295235191188 \end{aligned}\]

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 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*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 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*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 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*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^20 - 7*x^19 + 19*x^18 - 5*x^17 - 108*x^16 + 311*x^15 - 333*x^14 - 167*x^13 + 992*x^12 - 1203*x^11 + 141*x^10 + 1555*x^9 - 2554*x^8 + 2293*x^7 - 1387*x^6 + 609*x^5 - 208*x^4 + 77*x^3 - 33*x^2 + 9*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

$S_5$ (as 20T35):

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 non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

10.4.133541229312.1

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 5 sibling: 5.3.55728.1
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.3.55728.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{5}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.3.0.1}{3} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.4.5.16a1.2$x^{20} + 5 x^{17} + 5 x^{16} + 10 x^{14} + 20 x^{13} + 10 x^{12} + 10 x^{11} + 30 x^{10} + 30 x^{9} + 15 x^{8} + 20 x^{7} + 30 x^{6} + 21 x^{5} + 10 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 3$$5$$4$$16$20T5$$[\ ]_{5}^{4}$$
\(3\) Copy content Toggle raw display 3.4.5.16a1.2$x^{20} + 10 x^{19} + 40 x^{18} + 80 x^{17} + 90 x^{16} + 112 x^{15} + 240 x^{14} + 320 x^{13} + 200 x^{12} + 240 x^{11} + 480 x^{10} + 320 x^{9} + 80 x^{8} + 320 x^{7} + 320 x^{6} + 80 x^{4} + 160 x^{3} + 35$$5$$4$$16$20T5$$[\ ]_{5}^{4}$$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
43.3.2.3a1.2$x^{6} + 2 x^{4} + 80 x^{3} + x^{2} + 80 x + 1643$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
43.3.2.3a1.2$x^{6} + 2 x^{4} + 80 x^{3} + x^{2} + 80 x + 1643$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
43.3.2.3a1.2$x^{6} + 2 x^{4} + 80 x^{3} + x^{2} + 80 x + 1643$$2$$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)