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Properties

Label 16.8.220...125.6
Degree $16$
Signature $[8, 4]$
Discriminant $2.201\times 10^{30}$
Root discriminant \(78.78\)
Ramified primes $5,101$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^6.\SD_{16}$ (as 16T1250)

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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 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 12*y^14 + 14*y^13 - 89*y^12 + 152*y^11 + 651*y^10 + 332*y^9 - 821*y^8 - 6220*y^7 + 557*y^6 + 4897*y^5 + 7321*y^4 - 28007*y^3 - 78580*y^2 - 6439*y + 18769, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769)
 

\( x^{16} - x^{15} - 12 x^{14} + 14 x^{13} - 89 x^{12} + 152 x^{11} + 651 x^{10} + 332 x^{9} - 821 x^{8} + \cdots + 18769 \) 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:   \(2200830136976503360666408203125\) \(\medspace = 5^{9}\cdot 101^{12}\) 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:  \(78.78\)
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:  $5^{3/4}101^{3/4}\approx 106.52916761335055$
Ramified primes:   \(5\), \(101\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{23\cdots 95}a^{15}-\frac{33\cdots 14}{46\cdots 59}a^{14}+\frac{34\cdots 59}{46\cdots 59}a^{13}-\frac{33\cdots 92}{23\cdots 95}a^{12}+\frac{54\cdots 13}{23\cdots 95}a^{11}-\frac{64\cdots 18}{23\cdots 95}a^{10}+\frac{80\cdots 08}{23\cdots 95}a^{9}+\frac{15\cdots 84}{46\cdots 59}a^{8}-\frac{10\cdots 34}{23\cdots 95}a^{7}+\frac{98\cdots 51}{23\cdots 95}a^{6}+\frac{19\cdots 27}{23\cdots 95}a^{5}-\frac{10\cdots 99}{23\cdots 95}a^{4}+\frac{25\cdots 54}{23\cdots 95}a^{3}+\frac{51\cdots 48}{23\cdots 95}a^{2}-\frac{47\cdots 86}{23\cdots 95}a+\frac{14\cdots 56}{17\cdots 35}$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{55\cdots 23}{17\cdots 35}a^{15}+\frac{22\cdots 12}{34\cdots 07}a^{14}-\frac{36\cdots 06}{17\cdots 35}a^{13}-\frac{42\cdots 24}{17\cdots 35}a^{12}-\frac{57\cdots 76}{17\cdots 35}a^{11}-\frac{16\cdots 53}{34\cdots 07}a^{10}+\frac{11\cdots 83}{17\cdots 35}a^{9}+\frac{65\cdots 99}{17\cdots 35}a^{8}+\frac{11\cdots 76}{17\cdots 35}a^{7}-\frac{16\cdots 39}{17\cdots 35}a^{6}-\frac{90\cdots 88}{17\cdots 35}a^{5}+\frac{24\cdots 88}{17\cdots 35}a^{4}+\frac{14\cdots 81}{17\cdots 35}a^{3}+\frac{23\cdots 27}{34\cdots 07}a^{2}-\frac{36\cdots 18}{34\cdots 07}a+\frac{57\cdots 53}{17\cdots 35}$, $\frac{45\cdots 61}{17\cdots 35}a^{15}-\frac{34\cdots 43}{34\cdots 07}a^{14}-\frac{47\cdots 27}{17\cdots 35}a^{13}+\frac{21\cdots 67}{17\cdots 35}a^{12}-\frac{58\cdots 02}{17\cdots 35}a^{11}+\frac{34\cdots 89}{34\cdots 07}a^{10}+\frac{20\cdots 21}{17\cdots 35}a^{9}-\frac{72\cdots 47}{17\cdots 35}a^{8}-\frac{48\cdots 63}{17\cdots 35}a^{7}-\frac{18\cdots 48}{17\cdots 35}a^{6}+\frac{60\cdots 89}{17\cdots 35}a^{5}-\frac{58\cdots 84}{17\cdots 35}a^{4}-\frac{14\cdots 13}{17\cdots 35}a^{3}-\frac{76\cdots 02}{34\cdots 07}a^{2}+\frac{55\cdots 48}{34\cdots 07}a+\frac{22\cdots 66}{17\cdots 35}$, $\frac{36\cdots 38}{17\cdots 35}a^{15}+\frac{28\cdots 33}{34\cdots 07}a^{14}-\frac{23\cdots 26}{17\cdots 35}a^{13}+\frac{14\cdots 06}{17\cdots 35}a^{12}-\frac{41\cdots 56}{17\cdots 35}a^{11}+\frac{34\cdots 38}{34\cdots 07}a^{10}+\frac{14\cdots 43}{17\cdots 35}a^{9}+\frac{31\cdots 89}{17\cdots 35}a^{8}+\frac{26\cdots 71}{17\cdots 35}a^{7}-\frac{81\cdots 79}{17\cdots 35}a^{6}+\frac{21\cdots 42}{17\cdots 35}a^{5}+\frac{99\cdots 23}{17\cdots 35}a^{4}+\frac{15\cdots 71}{17\cdots 35}a^{3}-\frac{21\cdots 49}{34\cdots 07}a^{2}-\frac{13\cdots 68}{34\cdots 07}a+\frac{57\cdots 28}{17\cdots 35}$, $\frac{32\cdots 05}{46\cdots 59}a^{15}+\frac{15\cdots 29}{46\cdots 59}a^{14}-\frac{40\cdots 09}{46\cdots 59}a^{13}-\frac{13\cdots 36}{46\cdots 59}a^{12}-\frac{28\cdots 01}{46\cdots 59}a^{11}+\frac{56\cdots 10}{46\cdots 59}a^{10}+\frac{27\cdots 71}{46\cdots 59}a^{9}+\frac{50\cdots 05}{46\cdots 59}a^{8}+\frac{27\cdots 86}{46\cdots 59}a^{7}-\frac{19\cdots 09}{46\cdots 59}a^{6}-\frac{40\cdots 61}{46\cdots 59}a^{5}-\frac{48\cdots 63}{46\cdots 59}a^{4}-\frac{11\cdots 59}{46\cdots 59}a^{3}+\frac{33\cdots 47}{46\cdots 59}a^{2}-\frac{47\cdots 83}{46\cdots 59}a-\frac{34\cdots 22}{34\cdots 07}$, $\frac{14\cdots 26}{23\cdots 95}a^{15}+\frac{10\cdots 86}{23\cdots 95}a^{14}-\frac{11\cdots 08}{23\cdots 95}a^{13}+\frac{80\cdots 00}{46\cdots 59}a^{12}-\frac{15\cdots 61}{23\cdots 95}a^{11}-\frac{36\cdots 38}{23\cdots 95}a^{10}+\frac{45\cdots 33}{23\cdots 95}a^{9}+\frac{11\cdots 46}{23\cdots 95}a^{8}+\frac{16\cdots 86}{23\cdots 95}a^{7}-\frac{25\cdots 67}{23\cdots 95}a^{6}+\frac{22\cdots 73}{23\cdots 95}a^{5}+\frac{38\cdots 32}{23\cdots 95}a^{4}+\frac{26\cdots 52}{23\cdots 95}a^{3}+\frac{11\cdots 26}{23\cdots 95}a^{2}-\frac{15\cdots 10}{46\cdots 59}a-\frac{39\cdots 63}{17\cdots 35}$, $\frac{16\cdots 26}{23\cdots 95}a^{15}+\frac{59\cdots 27}{23\cdots 95}a^{14}-\frac{93\cdots 84}{23\cdots 95}a^{13}+\frac{69\cdots 58}{23\cdots 95}a^{12}-\frac{37\cdots 22}{46\cdots 59}a^{11}+\frac{62\cdots 79}{23\cdots 95}a^{10}+\frac{58\cdots 25}{46\cdots 59}a^{9}+\frac{16\cdots 09}{23\cdots 95}a^{8}+\frac{19\cdots 25}{46\cdots 59}a^{7}+\frac{25\cdots 14}{23\cdots 95}a^{6}+\frac{72\cdots 77}{23\cdots 95}a^{5}+\frac{62\cdots 83}{23\cdots 95}a^{4}+\frac{20\cdots 07}{23\cdots 95}a^{3}+\frac{12\cdots 02}{23\cdots 95}a^{2}+\frac{42\cdots 49}{46\cdots 59}a-\frac{21\cdots 39}{17\cdots 35}$, $\frac{23\cdots 62}{46\cdots 59}a^{15}-\frac{74\cdots 20}{46\cdots 59}a^{14}-\frac{15\cdots 21}{23\cdots 95}a^{13}+\frac{29\cdots 57}{23\cdots 95}a^{12}-\frac{23\cdots 31}{46\cdots 59}a^{11}+\frac{53\cdots 79}{23\cdots 95}a^{10}+\frac{69\cdots 19}{23\cdots 95}a^{9}+\frac{26\cdots 69}{23\cdots 95}a^{8}-\frac{28\cdots 17}{23\cdots 95}a^{7}-\frac{14\cdots 47}{23\cdots 95}a^{6}+\frac{44\cdots 26}{23\cdots 95}a^{5}+\frac{16\cdots 05}{46\cdots 59}a^{4}+\frac{15\cdots 34}{23\cdots 95}a^{3}-\frac{11\cdots 89}{23\cdots 95}a^{2}-\frac{13\cdots 27}{23\cdots 95}a+\frac{55\cdots 46}{17\cdots 35}$, $\frac{32\cdots 16}{46\cdots 59}a^{15}-\frac{42\cdots 37}{23\cdots 95}a^{14}-\frac{20\cdots 01}{46\cdots 59}a^{13}+\frac{15\cdots 13}{23\cdots 95}a^{12}-\frac{37\cdots 92}{23\cdots 95}a^{11}+\frac{40\cdots 29}{23\cdots 95}a^{10}+\frac{65\cdots 94}{23\cdots 95}a^{9}+\frac{15\cdots 92}{23\cdots 95}a^{8}-\frac{81\cdots 82}{23\cdots 95}a^{7}-\frac{33\cdots 76}{23\cdots 95}a^{6}+\frac{79\cdots 79}{23\cdots 95}a^{5}+\frac{39\cdots 68}{23\cdots 95}a^{4}-\frac{25\cdots 17}{23\cdots 95}a^{3}-\frac{10\cdots 86}{46\cdots 59}a^{2}-\frac{57\cdots 09}{23\cdots 95}a+\frac{23\cdots 59}{17\cdots 35}$, $\frac{45\cdots 13}{23\cdots 95}a^{15}-\frac{51\cdots 11}{23\cdots 95}a^{14}-\frac{56\cdots 64}{23\cdots 95}a^{13}+\frac{73\cdots 51}{23\cdots 95}a^{12}-\frac{38\cdots 22}{23\cdots 95}a^{11}+\frac{68\cdots 74}{23\cdots 95}a^{10}+\frac{29\cdots 52}{23\cdots 95}a^{9}+\frac{12\cdots 52}{23\cdots 95}a^{8}-\frac{72\cdots 11}{23\cdots 95}a^{7}-\frac{24\cdots 78}{23\cdots 95}a^{6}+\frac{13\cdots 62}{23\cdots 95}a^{5}+\frac{16\cdots 97}{23\cdots 95}a^{4}+\frac{51\cdots 07}{23\cdots 95}a^{3}-\frac{18\cdots 62}{23\cdots 95}a^{2}-\frac{29\cdots 88}{23\cdots 95}a+\frac{12\cdots 44}{17\cdots 35}$, $\frac{12\cdots 33}{23\cdots 95}a^{15}+\frac{10\cdots 77}{23\cdots 95}a^{14}-\frac{10\cdots 16}{23\cdots 95}a^{13}+\frac{13\cdots 43}{23\cdots 95}a^{12}-\frac{11\cdots 54}{23\cdots 95}a^{11}-\frac{42\cdots 04}{23\cdots 95}a^{10}+\frac{41\cdots 69}{23\cdots 95}a^{9}+\frac{11\cdots 37}{23\cdots 95}a^{8}+\frac{99\cdots 88}{23\cdots 95}a^{7}-\frac{24\cdots 43}{23\cdots 95}a^{6}+\frac{16\cdots 88}{23\cdots 95}a^{5}+\frac{40\cdots 98}{46\cdots 59}a^{4}+\frac{23\cdots 38}{23\cdots 95}a^{3}+\frac{11\cdots 65}{46\cdots 59}a^{2}-\frac{30\cdots 11}{23\cdots 95}a-\frac{22\cdots 35}{34\cdots 07}$, $\frac{11\cdots 20}{46\cdots 59}a^{15}+\frac{12\cdots 59}{23\cdots 95}a^{14}-\frac{59\cdots 13}{23\cdots 95}a^{13}+\frac{45\cdots 13}{46\cdots 59}a^{12}-\frac{51\cdots 96}{23\cdots 95}a^{11}+\frac{16\cdots 39}{23\cdots 95}a^{10}+\frac{31\cdots 04}{23\cdots 95}a^{9}+\frac{43\cdots 98}{23\cdots 95}a^{8}+\frac{20\cdots 68}{23\cdots 95}a^{7}-\frac{25\cdots 59}{23\cdots 95}a^{6}-\frac{17\cdots 02}{46\cdots 59}a^{5}+\frac{23\cdots 54}{23\cdots 95}a^{4}+\frac{72\cdots 21}{23\cdots 95}a^{3}-\frac{38\cdots 77}{23\cdots 95}a^{2}-\frac{33\cdots 23}{23\cdots 95}a-\frac{26\cdots 68}{34\cdots 07}$ 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:  \( 2150910676.85 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 2150910676.85 \cdot 2}{2\cdot\sqrt{2200830136976503360666408203125}}\cr\approx \mathstrut & 0.578480270882 \end{aligned}\] (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 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769) 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 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769, 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 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769); 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^16 - x^15 - 12*x^14 + 14*x^13 - 89*x^12 + 152*x^11 + 651*x^10 + 332*x^9 - 821*x^8 - 6220*x^7 + 557*x^6 + 4897*x^5 + 7321*x^4 - 28007*x^3 - 78580*x^2 - 6439*x + 18769); 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

$C_2^6.\SD_{16}$ (as 16T1250):

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 1024
The 34 conjugacy class representatives for $C_2^6.\SD_{16}$
Character table for $C_2^6.\SD_{16}$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 8.8.132690018825125.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.88033205479060134426656328125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ $16$ R $16$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(101\) Copy content Toggle raw display 101.1.4.3a1.3$x^{4} + 404$$4$$1$$3$$C_4$$$[\ ]_{4}$$
101.1.4.3a1.3$x^{4} + 404$$4$$1$$3$$C_4$$$[\ ]_{4}$$
101.2.4.6a1.2$x^{8} + 388 x^{7} + 56462 x^{6} + 3653020 x^{5} + 88755121 x^{4} + 7306040 x^{3} + 225848 x^{2} + 3104 x + 117$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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