↑

Properties

Label 16.8.336...609.3
Degree $16$
Signature $[8, 4]$
Discriminant $3.363\times 10^{38}$
Root discriminant \(255.81\)
Ramified primes $61,97$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^6:C_8$ (as 16T817)

Related objects

Downloads

Learn more

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 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 + 114*y^14 + 512*y^13 + 1901*y^12 - 8920*y^11 - 171530*y^10 + 412377*y^9 + 1231086*y^8 - 275150*y^7 - 1736631*y^6 - 52474780*y^5 + 58963430*y^4 + 195632765*y^3 - 399339032*y^2 + 247945353*y - 49669687, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687)
 

\( x^{16} - 2 x^{15} + 114 x^{14} + 512 x^{13} + 1901 x^{12} - 8920 x^{11} - 171530 x^{10} + \cdots - 49669687 \) 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:   \(336343694112121707284133254922278323609\) \(\medspace = 61^{6}\cdot 97^{14}\) 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:  \(255.81\)
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:  $61^{1/2}97^{7/8}\approx 427.6516824854611$
Ramified primes:   \(61\), \(97\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{11}-\frac{1}{6}a^{10}+\frac{1}{4}a^{9}-\frac{1}{6}a^{8}-\frac{1}{4}a^{6}+\frac{1}{3}a^{4}+\frac{1}{12}a^{3}+\frac{1}{3}a^{2}+\frac{1}{12}$, $\frac{1}{732}a^{13}+\frac{5}{366}a^{12}-\frac{1}{6}a^{11}-\frac{83}{244}a^{10}-\frac{121}{366}a^{9}+\frac{17}{61}a^{8}+\frac{103}{244}a^{7}-\frac{24}{61}a^{6}-\frac{56}{183}a^{5}+\frac{37}{732}a^{4}+\frac{19}{183}a^{3}-\frac{8}{61}a^{2}+\frac{301}{732}a-\frac{19}{61}$, $\frac{1}{732}a^{14}+\frac{11}{366}a^{12}-\frac{83}{244}a^{11}+\frac{74}{183}a^{10}-\frac{76}{183}a^{9}-\frac{23}{732}a^{8}+\frac{47}{122}a^{7}-\frac{68}{183}a^{6}+\frac{27}{244}a^{5}-\frac{25}{366}a^{4}+\frac{10}{61}a^{3}+\frac{41}{732}a^{2}-\frac{155}{366}a+\frac{82}{183}$, $\frac{1}{49\cdots 76}a^{15}+\frac{87\cdots 95}{16\cdots 92}a^{14}+\frac{50\cdots 75}{12\cdots 19}a^{13}+\frac{21\cdots 35}{12\cdots 19}a^{12}-\frac{18\cdots 55}{49\cdots 76}a^{11}-\frac{73\cdots 10}{41\cdots 73}a^{10}+\frac{88\cdots 89}{24\cdots 38}a^{9}+\frac{10\cdots 25}{49\cdots 76}a^{8}-\frac{10\cdots 39}{12\cdots 19}a^{7}-\frac{13\cdots 36}{41\cdots 73}a^{6}-\frac{22\cdots 57}{49\cdots 76}a^{5}+\frac{58\cdots 53}{12\cdots 19}a^{4}+\frac{52\cdots 12}{12\cdots 19}a^{3}+\frac{61\cdots 09}{16\cdots 92}a^{2}-\frac{24\cdots 28}{12\cdots 19}a+\frac{14\cdots 89}{11\cdots 32}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (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:   $\frac{81\cdots 97}{49\cdots 76}a^{15}-\frac{89\cdots 73}{49\cdots 76}a^{14}+\frac{46\cdots 79}{24\cdots 38}a^{13}+\frac{16\cdots 01}{16\cdots 92}a^{12}+\frac{20\cdots 25}{49\cdots 76}a^{11}-\frac{45\cdots 11}{41\cdots 73}a^{10}-\frac{48\cdots 35}{16\cdots 92}a^{9}+\frac{68\cdots 27}{16\cdots 92}a^{8}+\frac{59\cdots 87}{24\cdots 38}a^{7}+\frac{86\cdots 55}{49\cdots 76}a^{6}-\frac{65\cdots 47}{49\cdots 76}a^{5}-\frac{72\cdots 15}{83\cdots 46}a^{4}+\frac{87\cdots 31}{49\cdots 76}a^{3}+\frac{56\cdots 39}{16\cdots 92}a^{2}-\frac{28\cdots 09}{83\cdots 46}a+\frac{16\cdots 27}{19\cdots 22}$, $\frac{87\cdots 99}{49\cdots 76}a^{15}-\frac{17\cdots 57}{49\cdots 76}a^{14}+\frac{16\cdots 85}{83\cdots 46}a^{13}+\frac{44\cdots 03}{49\cdots 76}a^{12}+\frac{14\cdots 17}{49\cdots 76}a^{11}-\frac{21\cdots 04}{12\cdots 19}a^{10}-\frac{15\cdots 97}{49\cdots 76}a^{9}+\frac{12\cdots 47}{16\cdots 92}a^{8}+\frac{68\cdots 77}{24\cdots 38}a^{7}-\frac{37\cdots 57}{49\cdots 76}a^{6}-\frac{11\cdots 45}{16\cdots 92}a^{5}-\frac{24\cdots 53}{24\cdots 38}a^{4}+\frac{50\cdots 11}{49\cdots 76}a^{3}+\frac{40\cdots 21}{81\cdots 16}a^{2}-\frac{15\cdots 81}{24\cdots 38}a-\frac{66\cdots 21}{96\cdots 11}$, $\frac{16\cdots 63}{84\cdots 18}a^{15}-\frac{53\cdots 37}{25\cdots 54}a^{14}+\frac{28\cdots 78}{12\cdots 27}a^{13}+\frac{50\cdots 49}{42\cdots 09}a^{12}+\frac{12\cdots 55}{25\cdots 54}a^{11}-\frac{16\cdots 95}{12\cdots 27}a^{10}-\frac{44\cdots 88}{12\cdots 27}a^{9}+\frac{12\cdots 63}{25\cdots 54}a^{8}+\frac{12\cdots 44}{42\cdots 09}a^{7}+\frac{26\cdots 27}{12\cdots 27}a^{6}-\frac{36\cdots 49}{25\cdots 54}a^{5}-\frac{43\cdots 27}{42\cdots 09}a^{4}+\frac{23\cdots 73}{12\cdots 27}a^{3}+\frac{10\cdots 17}{25\cdots 54}a^{2}-\frac{51\cdots 25}{12\cdots 27}a+\frac{26\cdots 31}{25\cdots 54}$, $\frac{51\cdots 13}{40\cdots 58}a^{15}-\frac{23\cdots 31}{16\cdots 92}a^{14}+\frac{35\cdots 69}{24\cdots 38}a^{13}+\frac{64\cdots 03}{83\cdots 46}a^{12}+\frac{15\cdots 25}{49\cdots 76}a^{11}-\frac{21\cdots 11}{24\cdots 38}a^{10}-\frac{28\cdots 83}{12\cdots 19}a^{9}+\frac{16\cdots 59}{49\cdots 76}a^{8}+\frac{23\cdots 94}{12\cdots 19}a^{7}+\frac{52\cdots 76}{41\cdots 73}a^{6}-\frac{57\cdots 89}{49\cdots 76}a^{5}-\frac{28\cdots 79}{41\cdots 73}a^{4}+\frac{19\cdots 35}{12\cdots 19}a^{3}+\frac{13\cdots 27}{49\cdots 76}a^{2}-\frac{34\cdots 77}{12\cdots 19}a+\frac{37\cdots 15}{57\cdots 66}$, $\frac{74\cdots 38}{28\cdots 33}a^{15}-\frac{10\cdots 69}{38\cdots 44}a^{14}+\frac{16\cdots 49}{57\cdots 66}a^{13}+\frac{45\cdots 24}{28\cdots 33}a^{12}+\frac{72\cdots 19}{11\cdots 32}a^{11}-\frac{33\cdots 71}{19\cdots 22}a^{10}-\frac{26\cdots 71}{57\cdots 66}a^{9}+\frac{75\cdots 01}{11\cdots 32}a^{8}+\frac{10\cdots 78}{28\cdots 33}a^{7}+\frac{50\cdots 57}{19\cdots 22}a^{6}-\frac{24\cdots 31}{11\cdots 32}a^{5}-\frac{39\cdots 35}{28\cdots 33}a^{4}+\frac{17\cdots 45}{57\cdots 66}a^{3}+\frac{20\cdots 19}{38\cdots 44}a^{2}-\frac{15\cdots 22}{28\cdots 33}a+\frac{77\cdots 15}{57\cdots 66}$, $\frac{39\cdots 81}{16\cdots 36}a^{15}-\frac{11\cdots 35}{50\cdots 08}a^{14}+\frac{66\cdots 81}{25\cdots 54}a^{13}+\frac{12\cdots 15}{84\cdots 18}a^{12}+\frac{29\cdots 49}{50\cdots 08}a^{11}-\frac{37\cdots 67}{25\cdots 54}a^{10}-\frac{52\cdots 36}{12\cdots 27}a^{9}+\frac{27\cdots 13}{50\cdots 08}a^{8}+\frac{28\cdots 45}{84\cdots 18}a^{7}+\frac{69\cdots 13}{25\cdots 54}a^{6}-\frac{66\cdots 65}{50\cdots 08}a^{5}-\frac{10\cdots 33}{84\cdots 18}a^{4}+\frac{42\cdots 43}{25\cdots 54}a^{3}+\frac{23\cdots 19}{50\cdots 08}a^{2}-\frac{11\cdots 73}{25\cdots 54}a+\frac{97\cdots 59}{83\cdots 28}$, $\frac{44\cdots 63}{16\cdots 92}a^{15}-\frac{36\cdots 92}{12\cdots 19}a^{14}+\frac{37\cdots 36}{12\cdots 19}a^{13}+\frac{20\cdots 75}{12\cdots 19}a^{12}+\frac{54\cdots 31}{83\cdots 46}a^{11}-\frac{74\cdots 35}{41\cdots 73}a^{10}-\frac{58\cdots 15}{12\cdots 19}a^{9}+\frac{83\cdots 61}{12\cdots 19}a^{8}+\frac{32\cdots 29}{83\cdots 46}a^{7}+\frac{69\cdots 69}{24\cdots 38}a^{6}-\frac{53\cdots 97}{24\cdots 38}a^{5}-\frac{35\cdots 17}{24\cdots 38}a^{4}+\frac{23\cdots 75}{83\cdots 46}a^{3}+\frac{45\cdots 01}{83\cdots 46}a^{2}-\frac{14\cdots 25}{24\cdots 38}a+\frac{16\cdots 91}{11\cdots 32}$, $\frac{69\cdots 07}{83\cdots 46}a^{15}-\frac{10\cdots 23}{24\cdots 38}a^{14}+\frac{89\cdots 77}{83\cdots 46}a^{13}+\frac{46\cdots 40}{41\cdots 73}a^{12}+\frac{30\cdots 17}{24\cdots 38}a^{11}-\frac{27\cdots 95}{24\cdots 38}a^{10}-\frac{13\cdots 02}{12\cdots 19}a^{9}+\frac{56\cdots 97}{83\cdots 46}a^{8}-\frac{78\cdots 75}{83\cdots 46}a^{7}+\frac{30\cdots 44}{12\cdots 19}a^{6}-\frac{72\cdots 65}{83\cdots 46}a^{5}-\frac{15\cdots 07}{83\cdots 46}a^{4}+\frac{13\cdots 88}{12\cdots 19}a^{3}-\frac{35\cdots 79}{24\cdots 38}a^{2}+\frac{19\cdots 35}{24\cdots 38}a-\frac{28\cdots 51}{19\cdots 22}$, $\frac{19\cdots 95}{16\cdots 92}a^{15}-\frac{94\cdots 25}{12\cdots 19}a^{14}+\frac{76\cdots 95}{49\cdots 76}a^{13}-\frac{12\cdots 95}{41\cdots 73}a^{12}+\frac{52\cdots 49}{12\cdots 19}a^{11}-\frac{11\cdots 27}{49\cdots 76}a^{10}-\frac{20\cdots 99}{12\cdots 19}a^{9}+\frac{15\cdots 59}{12\cdots 19}a^{8}-\frac{24\cdots 61}{16\cdots 92}a^{7}+\frac{16\cdots 61}{12\cdots 19}a^{6}-\frac{29\cdots 41}{24\cdots 38}a^{5}-\frac{83\cdots 61}{16\cdots 92}a^{4}+\frac{31\cdots 25}{12\cdots 19}a^{3}-\frac{87\cdots 99}{24\cdots 38}a^{2}+\frac{96\cdots 97}{49\cdots 76}a-\frac{42\cdots 47}{11\cdots 32}$, $\frac{22\cdots 47}{83\cdots 46}a^{15}-\frac{11\cdots 31}{83\cdots 46}a^{14}+\frac{14\cdots 44}{41\cdots 73}a^{13}+\frac{28\cdots 59}{83\cdots 46}a^{12}+\frac{33\cdots 79}{83\cdots 46}a^{11}-\frac{15\cdots 52}{41\cdots 73}a^{10}-\frac{29\cdots 67}{83\cdots 46}a^{9}+\frac{18\cdots 93}{83\cdots 46}a^{8}-\frac{13\cdots 13}{41\cdots 73}a^{7}+\frac{67\cdots 87}{83\cdots 46}a^{6}-\frac{23\cdots 51}{83\cdots 46}a^{5}-\frac{24\cdots 22}{41\cdots 73}a^{4}+\frac{28\cdots 51}{83\cdots 46}a^{3}-\frac{40\cdots 43}{83\cdots 46}a^{2}+\frac{11\cdots 08}{41\cdots 73}a-\frac{53\cdots 98}{96\cdots 11}$, $\frac{33\cdots 39}{49\cdots 76}a^{15}-\frac{43\cdots 07}{41\cdots 73}a^{14}+\frac{37\cdots 81}{49\cdots 76}a^{13}+\frac{62\cdots 75}{16\cdots 92}a^{12}+\frac{35\cdots 61}{24\cdots 38}a^{11}-\frac{26\cdots 55}{49\cdots 76}a^{10}-\frac{58\cdots 81}{49\cdots 76}a^{9}+\frac{55\cdots 07}{24\cdots 38}a^{8}+\frac{45\cdots 79}{49\cdots 76}a^{7}+\frac{35\cdots 07}{16\cdots 92}a^{6}-\frac{26\cdots 63}{24\cdots 38}a^{5}-\frac{58\cdots 15}{16\cdots 92}a^{4}+\frac{11\cdots 89}{49\cdots 76}a^{3}+\frac{35\cdots 49}{24\cdots 38}a^{2}-\frac{10\cdots 53}{49\cdots 76}a+\frac{44\cdots 27}{57\cdots 66}$ 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:  \( 9467964682180 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 9467964682180 \cdot 2}{2\cdot\sqrt{336343694112121707284133254922278323609}}\cr\approx \mathstrut & 0.205979700855647 \end{aligned}\] (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 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687) 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 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687, 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 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687); 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 - 2*x^15 + 114*x^14 + 512*x^13 + 1901*x^12 - 8920*x^11 - 171530*x^10 + 412377*x^9 + 1231086*x^8 - 275150*x^7 - 1736631*x^6 - 52474780*x^5 + 58963430*x^4 + 195632765*x^3 - 399339032*x^2 + 247945353*x - 49669687); 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

$C_2^6:C_8$ (as 16T817):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 512
The 32 conjugacy class representatives for $C_2^6:C_8$
Character table for $C_2^6:C_8$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.24292037884309209334711647701449.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(61\) Copy content Toggle raw display $\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{61}$$x + 59$$1$$1$$0$Trivial$$[\ ]$$
61.1.2.1a1.1$x^{2} + 61$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.1.2.1a1.2$x^{2} + 122$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.1.2.1a1.2$x^{2} + 122$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.1.2.1a1.2$x^{2} + 122$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.1.2.1a1.2$x^{2} + 122$$2$$1$$1$$C_2$$$[\ ]_{2}$$
61.1.2.1a1.1$x^{2} + 61$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(97\) Copy content Toggle raw display 97.1.8.7a1.1$x^{8} + 97$$8$$1$$7$$C_8$$$[\ ]_{8}$$
97.1.8.7a1.1$x^{8} + 97$$8$$1$$7$$C_8$$$[\ ]_{8}$$

Spectrum of ring of integers

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