Properties

Label 16.8.810...016.3
Degree $16$
Signature $(8, 4)$
Discriminant $8.103\times 10^{25}$
Root discriminant \(41.62\)
Ramified primes $2,3,23$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^4:Q_8$ (as 16T333)

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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^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382)
 
Copy content gp:K = bnfinit(y^16 - 8*y^14 - 8*y^13 + 40*y^12 - 24*y^11 - 56*y^10 + 96*y^9 + 982*y^8 - 4624*y^7 + 7024*y^6 + 544*y^5 - 10756*y^4 + 8128*y^3 - 152*y^2 - 1568*y + 382, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382)
 

\( x^{16} - 8 x^{14} - 8 x^{13} + 40 x^{12} - 24 x^{11} - 56 x^{10} + 96 x^{9} + 982 x^{8} - 4624 x^{7} + \cdots + 382 \) 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:  $16$
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:  $(8, 4)$
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:   \(81030865406861733858902016\) \(\medspace = 2^{58}\cdot 3^{12}\cdot 23^{2}\) 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:  \(41.62\)
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:  $2^{31/8}3^{3/4}23^{1/2}\approx 160.39694500305947$
Ramified primes:   \(2\), \(3\), \(23\) 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: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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{2}{7}a^{10}+\frac{3}{7}a^{9}+\frac{2}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{49}a^{14}+\frac{3}{49}a^{13}+\frac{11}{49}a^{12}+\frac{6}{49}a^{11}+\frac{3}{7}a^{10}+\frac{1}{49}a^{9}+\frac{10}{49}a^{8}-\frac{11}{49}a^{7}+\frac{20}{49}a^{6}-\frac{19}{49}a^{5}+\frac{13}{49}a^{4}+\frac{1}{49}a^{3}+\frac{10}{49}a^{2}-\frac{15}{49}a+\frac{1}{49}$, $\frac{1}{36\cdots 03}a^{15}-\frac{43\cdots 35}{36\cdots 03}a^{14}-\frac{15\cdots 61}{36\cdots 03}a^{13}+\frac{97\cdots 22}{51\cdots 29}a^{12}-\frac{35\cdots 53}{36\cdots 03}a^{11}+\frac{16\cdots 61}{36\cdots 03}a^{10}-\frac{16\cdots 43}{36\cdots 03}a^{9}-\frac{24\cdots 27}{36\cdots 03}a^{8}+\frac{88\cdots 81}{36\cdots 03}a^{7}-\frac{17\cdots 86}{51\cdots 29}a^{6}-\frac{13\cdots 84}{36\cdots 03}a^{5}-\frac{10\cdots 05}{36\cdots 03}a^{4}+\frac{71\cdots 91}{36\cdots 03}a^{3}+\frac{56\cdots 49}{36\cdots 03}a^{2}-\frac{12\cdots 96}{36\cdots 03}a-\frac{50\cdots 06}{36\cdots 03}$ 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 $2$ (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:  $C_{2}\times C_{2}$, 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:  $11$
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:   $\frac{27\cdots 18}{51\cdots 29}a^{15}-\frac{40\cdots 98}{51\cdots 29}a^{14}-\frac{28\cdots 22}{51\cdots 29}a^{13}+\frac{28\cdots 79}{51\cdots 29}a^{12}+\frac{18\cdots 38}{51\cdots 29}a^{11}-\frac{11\cdots 10}{51\cdots 29}a^{10}-\frac{21\cdots 16}{51\cdots 29}a^{9}+\frac{45\cdots 72}{51\cdots 29}a^{8}+\frac{26\cdots 48}{51\cdots 29}a^{7}-\frac{16\cdots 74}{51\cdots 29}a^{6}+\frac{30\cdots 80}{51\cdots 29}a^{5}-\frac{25\cdots 09}{51\cdots 29}a^{4}-\frac{52\cdots 80}{51\cdots 29}a^{3}+\frac{35\cdots 34}{51\cdots 29}a^{2}+\frac{93\cdots 36}{51\cdots 29}a-\frac{67\cdots 97}{51\cdots 29}$, $\frac{59\cdots 64}{10\cdots 21}a^{15}+\frac{11\cdots 98}{10\cdots 21}a^{14}-\frac{36\cdots 28}{10\cdots 21}a^{13}-\frac{14\cdots 67}{10\cdots 21}a^{12}+\frac{72\cdots 36}{10\cdots 21}a^{11}+\frac{14\cdots 34}{10\cdots 21}a^{10}+\frac{20\cdots 84}{10\cdots 21}a^{9}+\frac{76\cdots 05}{10\cdots 21}a^{8}+\frac{63\cdots 72}{10\cdots 21}a^{7}-\frac{16\cdots 16}{10\cdots 21}a^{6}+\frac{15\cdots 72}{10\cdots 21}a^{5}+\frac{20\cdots 42}{10\cdots 21}a^{4}-\frac{15\cdots 04}{10\cdots 21}a^{3}-\frac{14\cdots 28}{10\cdots 21}a^{2}+\frac{28\cdots 00}{10\cdots 21}a-\frac{25\cdots 95}{10\cdots 21}$, $\frac{12\cdots 22}{51\cdots 29}a^{15}+\frac{15\cdots 16}{51\cdots 29}a^{14}+\frac{12\cdots 46}{51\cdots 29}a^{13}+\frac{90\cdots 44}{51\cdots 29}a^{12}-\frac{79\cdots 58}{51\cdots 29}a^{11}+\frac{45\cdots 16}{51\cdots 29}a^{10}+\frac{88\cdots 28}{51\cdots 29}a^{9}-\frac{19\cdots 96}{51\cdots 29}a^{8}-\frac{11\cdots 64}{51\cdots 29}a^{7}+\frac{72\cdots 22}{51\cdots 29}a^{6}-\frac{12\cdots 44}{51\cdots 29}a^{5}+\frac{74\cdots 99}{51\cdots 29}a^{4}+\frac{21\cdots 84}{51\cdots 29}a^{3}-\frac{14\cdots 70}{51\cdots 29}a^{2}-\frac{38\cdots 00}{51\cdots 29}a+\frac{35\cdots 33}{51\cdots 29}$, $\frac{19\cdots 11}{36\cdots 03}a^{15}-\frac{15\cdots 88}{36\cdots 03}a^{14}+\frac{14\cdots 14}{36\cdots 03}a^{13}+\frac{38\cdots 66}{51\cdots 29}a^{12}-\frac{58\cdots 66}{36\cdots 03}a^{11}+\frac{43\cdots 35}{36\cdots 03}a^{10}+\frac{11\cdots 68}{36\cdots 03}a^{9}-\frac{10\cdots 21}{36\cdots 03}a^{8}-\frac{20\cdots 16}{36\cdots 03}a^{7}+\frac{10\cdots 37}{51\cdots 29}a^{6}-\frac{82\cdots 92}{36\cdots 03}a^{5}-\frac{71\cdots 42}{36\cdots 03}a^{4}+\frac{15\cdots 88}{36\cdots 03}a^{3}-\frac{44\cdots 30}{36\cdots 03}a^{2}-\frac{27\cdots 80}{36\cdots 03}a+\frac{10\cdots 65}{36\cdots 03}$, $\frac{55\cdots 21}{36\cdots 03}a^{15}-\frac{81\cdots 66}{36\cdots 03}a^{14}+\frac{36\cdots 84}{36\cdots 03}a^{13}+\frac{14\cdots 45}{51\cdots 29}a^{12}-\frac{96\cdots 93}{36\cdots 03}a^{11}-\frac{61\cdots 22}{36\cdots 03}a^{10}+\frac{28\cdots 70}{36\cdots 03}a^{9}-\frac{94\cdots 62}{36\cdots 03}a^{8}-\frac{57\cdots 66}{36\cdots 03}a^{7}+\frac{25\cdots 62}{51\cdots 29}a^{6}-\frac{10\cdots 00}{36\cdots 03}a^{5}-\frac{28\cdots 21}{36\cdots 03}a^{4}+\frac{25\cdots 42}{36\cdots 03}a^{3}+\frac{73\cdots 49}{36\cdots 03}a^{2}-\frac{48\cdots 08}{36\cdots 03}a-\frac{97\cdots 05}{36\cdots 03}$, $\frac{66\cdots 69}{36\cdots 03}a^{15}-\frac{25\cdots 62}{36\cdots 03}a^{14}+\frac{51\cdots 70}{36\cdots 03}a^{13}+\frac{14\cdots 39}{74\cdots 47}a^{12}-\frac{23\cdots 16}{36\cdots 03}a^{11}+\frac{70\cdots 49}{36\cdots 03}a^{10}+\frac{39\cdots 32}{36\cdots 03}a^{9}-\frac{48\cdots 88}{36\cdots 03}a^{8}-\frac{66\cdots 20}{36\cdots 03}a^{7}+\frac{40\cdots 57}{51\cdots 29}a^{6}-\frac{35\cdots 00}{36\cdots 03}a^{5}-\frac{16\cdots 22}{36\cdots 03}a^{4}+\frac{64\cdots 44}{36\cdots 03}a^{3}-\frac{29\cdots 44}{36\cdots 03}a^{2}-\frac{98\cdots 60}{36\cdots 03}a+\frac{62\cdots 33}{36\cdots 03}$, $\frac{11\cdots 36}{36\cdots 03}a^{15}-\frac{11\cdots 40}{36\cdots 03}a^{14}+\frac{79\cdots 32}{36\cdots 03}a^{13}+\frac{24\cdots 74}{51\cdots 29}a^{12}-\frac{27\cdots 03}{36\cdots 03}a^{11}-\frac{22\cdots 85}{36\cdots 03}a^{10}+\frac{61\cdots 06}{36\cdots 03}a^{9}-\frac{44\cdots 78}{36\cdots 03}a^{8}-\frac{11\cdots 38}{36\cdots 03}a^{7}+\frac{57\cdots 97}{51\cdots 29}a^{6}-\frac{37\cdots 92}{36\cdots 03}a^{5}-\frac{45\cdots 36}{36\cdots 03}a^{4}+\frac{74\cdots 98}{36\cdots 03}a^{3}-\frac{12\cdots 05}{36\cdots 03}a^{2}-\frac{11\cdots 68}{36\cdots 03}a+\frac{35\cdots 03}{36\cdots 03}$, $\frac{25\cdots 84}{36\cdots 03}a^{15}-\frac{18\cdots 23}{36\cdots 03}a^{14}+\frac{19\cdots 41}{36\cdots 03}a^{13}+\frac{48\cdots 30}{51\cdots 29}a^{12}-\frac{78\cdots 44}{36\cdots 03}a^{11}+\frac{61\cdots 41}{36\cdots 03}a^{10}+\frac{14\cdots 21}{36\cdots 03}a^{9}-\frac{14\cdots 02}{36\cdots 03}a^{8}-\frac{26\cdots 92}{36\cdots 03}a^{7}+\frac{14\cdots 98}{51\cdots 29}a^{6}-\frac{10\cdots 34}{36\cdots 03}a^{5}-\frac{91\cdots 52}{36\cdots 03}a^{4}+\frac{21\cdots 50}{36\cdots 03}a^{3}-\frac{59\cdots 64}{36\cdots 03}a^{2}-\frac{37\cdots 03}{36\cdots 03}a+\frac{13\cdots 23}{36\cdots 03}$, $\frac{25\cdots 36}{36\cdots 03}a^{15}+\frac{18\cdots 37}{36\cdots 03}a^{14}-\frac{19\cdots 45}{36\cdots 03}a^{13}-\frac{48\cdots 13}{51\cdots 29}a^{12}+\frac{78\cdots 92}{36\cdots 03}a^{11}-\frac{60\cdots 79}{36\cdots 03}a^{10}-\frac{14\cdots 09}{36\cdots 03}a^{9}+\frac{14\cdots 17}{36\cdots 03}a^{8}+\frac{26\cdots 88}{36\cdots 03}a^{7}-\frac{14\cdots 82}{51\cdots 29}a^{6}+\frac{10\cdots 30}{36\cdots 03}a^{5}+\frac{91\cdots 58}{36\cdots 03}a^{4}-\frac{21\cdots 22}{36\cdots 03}a^{3}+\frac{59\cdots 60}{36\cdots 03}a^{2}+\frac{37\cdots 03}{36\cdots 03}a-\frac{13\cdots 05}{36\cdots 03}$, $\frac{72\cdots 73}{36\cdots 03}a^{15}+\frac{35\cdots 54}{36\cdots 03}a^{14}-\frac{56\cdots 12}{36\cdots 03}a^{13}-\frac{12\cdots 12}{51\cdots 29}a^{12}+\frac{24\cdots 07}{36\cdots 03}a^{11}-\frac{49\cdots 45}{36\cdots 03}a^{10}-\frac{43\cdots 45}{36\cdots 03}a^{9}+\frac{48\cdots 00}{36\cdots 03}a^{8}+\frac{73\cdots 53}{36\cdots 03}a^{7}-\frac{42\cdots 70}{51\cdots 29}a^{6}+\frac{36\cdots 78}{36\cdots 03}a^{5}+\frac{21\cdots 15}{36\cdots 03}a^{4}-\frac{67\cdots 00}{36\cdots 03}a^{3}+\frac{25\cdots 90}{36\cdots 03}a^{2}+\frac{11\cdots 06}{36\cdots 03}a-\frac{54\cdots 71}{36\cdots 03}$, $\frac{73\cdots 17}{36\cdots 03}a^{15}+\frac{73\cdots 11}{36\cdots 03}a^{14}-\frac{52\cdots 39}{36\cdots 03}a^{13}-\frac{15\cdots 99}{51\cdots 29}a^{12}+\frac{18\cdots 95}{36\cdots 03}a^{11}+\frac{15\cdots 52}{36\cdots 03}a^{10}-\frac{40\cdots 71}{36\cdots 03}a^{9}+\frac{30\cdots 09}{36\cdots 03}a^{8}+\frac{75\cdots 51}{36\cdots 03}a^{7}-\frac{38\cdots 87}{51\cdots 29}a^{6}+\frac{24\cdots 86}{36\cdots 03}a^{5}+\frac{30\cdots 89}{36\cdots 03}a^{4}-\frac{50\cdots 94}{36\cdots 03}a^{3}+\frac{80\cdots 81}{36\cdots 03}a^{2}+\frac{90\cdots 64}{36\cdots 03}a-\frac{26\cdots 81}{36\cdots 03}$ 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:  \( 6500801.080357627 \) (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)
 
Unit signature rank:  \( 7 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 6500801.080357627 \cdot 2}{2\cdot\sqrt{81030865406861733858902016}}\cr\approx \mathstrut & 0.288138358238721 \end{aligned}\] (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^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382) 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^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382, 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^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382); 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^16 - 8*x^14 - 8*x^13 + 40*x^12 - 24*x^11 - 56*x^10 + 96*x^9 + 982*x^8 - 4624*x^7 + 7024*x^6 + 544*x^5 - 10756*x^4 + 8128*x^3 - 152*x^2 - 1568*x + 382); 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

$C_2^4:Q_8$ (as 16T333):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 128
The 26 conjugacy class representatives for $C_2^4:Q_8$
Character table for $C_2^4:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{24})^+\), 8.4.562607161344.7, 8.8.12230590464.1, 8.4.62511906816.2

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 16 siblings: data not computed
Degree 32 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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.1.16.58n1.779$x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$$16$$1$$58$16T333$$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(23\) Copy content Toggle raw display 23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.1.2.1a1.1$x^{2} + 23$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.1.2.1a1.1$x^{2} + 23$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$

Spectrum of ring of integers

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