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Properties

Label 12.4.110...125.2
Degree $12$
Signature $[4, 4]$
Discriminant $1.108\times 10^{26}$
Root discriminant \(148.04\)
Ramified primes $5,197$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $A_5:C_4$ (as 12T124)

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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^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520)
 
Copy content gp:K = bnfinit(y^12 - 45*y^10 + 1050*y^8 - 13530*y^6 + 99685*y^4 - 328905*y^2 + 317520, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520)
 

\( x^{12} - 45x^{10} + 1050x^{8} - 13530x^{6} + 99685x^{4} - 328905x^{2} + 317520 \) 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:  $12$
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:  $[4, 4]$
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:   \(110764312692821648486328125\) \(\medspace = 5^{11}\cdot 197^{8}\) 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:  \(148.04\)
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:  not computed
Ramified primes:   \(5\), \(197\) 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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{5}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{6}-\frac{1}{8}a^{3}-\frac{1}{16}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{7}-\frac{1}{32}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{32}a^{3}+\frac{3}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{64}a^{8}+\frac{1}{64}a^{6}+\frac{3}{64}a^{4}-\frac{1}{8}a^{3}+\frac{11}{64}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{384}a^{9}-\frac{1}{128}a^{8}-\frac{1}{128}a^{7}+\frac{3}{128}a^{6}+\frac{1}{128}a^{5}-\frac{3}{128}a^{4}+\frac{5}{128}a^{3}+\frac{17}{128}a^{2}-\frac{5}{24}a+\frac{3}{8}$, $\frac{1}{969984}a^{10}-\frac{109}{26944}a^{8}+\frac{937}{161664}a^{6}-\frac{4771}{80832}a^{4}-\frac{1}{8}a^{3}+\frac{235729}{969984}a^{2}-\frac{3}{8}a-\frac{3133}{6736}$, $\frac{1}{40739328}a^{11}-\frac{1}{1939968}a^{10}+\frac{13}{47152}a^{9}-\frac{39}{6736}a^{8}+\frac{14929}{969984}a^{7}-\frac{3463}{323328}a^{6}-\frac{51011}{1697472}a^{5}+\frac{491}{80832}a^{4}+\frac{3433645}{40739328}a^{3}+\frac{82547}{1939968}a^{2}+\frac{16357}{94304}a-\frac{1919}{13472}$ 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:  $C_{4}$, 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:  $7$
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:   $\frac{69959}{5092416}a^{11}+\frac{14801}{484992}a^{10}+\frac{77843}{141456}a^{9}-\frac{2057}{1684}a^{8}-\frac{1419881}{121248}a^{7}+\frac{2099903}{80832}a^{6}+\frac{54416471}{424368}a^{5}-\frac{5746639}{20208}a^{4}-\frac{3764265671}{5092416}a^{3}+\frac{795000461}{484992}a^{2}+\frac{62813833}{70728}a-\frac{6639237}{3368}$, $\frac{45919}{40739328}a^{11}+\frac{2395}{1939968}a^{10}-\frac{81155}{1131648}a^{9}+\frac{4175}{53888}a^{8}+\frac{1511737}{969984}a^{7}-\frac{761825}{323328}a^{6}-\frac{65741245}{3394944}a^{5}+\frac{6722765}{161664}a^{4}+\frac{3405941695}{40739328}a^{3}-\frac{394046045}{1939968}a^{2}-\frac{25482395}{282912}a+\frac{3018097}{13472}$, $\frac{3883}{2909952}a^{11}-\frac{6515}{969984}a^{10}+\frac{3089}{80832}a^{9}+\frac{6223}{26944}a^{8}-\frac{293947}{484992}a^{7}-\frac{683759}{161664}a^{6}+\frac{826111}{242496}a^{5}+\frac{2998997}{80832}a^{4}+\frac{1056677}{2909952}a^{3}-\frac{135087227}{969984}a^{2}-\frac{561391}{20208}a+\frac{1136431}{6736}$, $\frac{3241673}{40739328}a^{11}+\frac{193691}{1939968}a^{10}-\frac{977563}{282912}a^{9}-\frac{116833}{26944}a^{8}+\frac{75759185}{969984}a^{7}+\frac{31689089}{323328}a^{6}-\frac{1618535557}{1697472}a^{5}-\frac{48356303}{40416}a^{4}+\frac{261930613733}{40739328}a^{3}+\frac{15650236199}{1939968}a^{2}-\frac{4538643997}{282912}a-\frac{271187443}{13472}$, $\frac{6497}{5092416}a^{11}+\frac{119}{53888}a^{10}-\frac{53875}{1131648}a^{9}-\frac{6803}{53888}a^{8}+\frac{530657}{484992}a^{7}+\frac{177711}{53888}a^{6}-\frac{48621787}{3394944}a^{5}-\frac{2286817}{53888}a^{4}+\frac{1114421605}{10184832}a^{3}+\frac{7504375}{26944}a^{2}-\frac{2830847}{8841}a-\frac{960969}{1684}$, $\frac{1576985}{6789888}a^{11}-\frac{12275}{40416}a^{10}-\frac{2849767}{282912}a^{9}+\frac{361685}{26944}a^{8}+\frac{36459305}{161664}a^{7}-\frac{8039657}{26944}a^{6}-\frac{385991015}{141456}a^{5}+\frac{96169275}{26944}a^{4}+\frac{123776675861}{6789888}a^{3}-\frac{1891518175}{80832}a^{2}-\frac{6396625013}{141456}a+\frac{96367723}{1684}$, $\frac{2545}{5092416}a^{11}+\frac{12509}{969984}a^{10}+\frac{78157}{1131648}a^{9}-\frac{7723}{53888}a^{8}-\frac{943031}{484992}a^{7}-\frac{86369}{80832}a^{6}+\frac{76897525}{3394944}a^{5}+\frac{6132065}{161664}a^{4}-\frac{941776915}{10184832}a^{3}-\frac{189671905}{969984}a^{2}+\frac{3416249}{35364}a+\frac{1473693}{6736}$ 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:  \( 851508807.1406436 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 851508807.1406436 \cdot 2}{2\cdot\sqrt{110764312692821648486328125}}\cr\approx \mathstrut & 2.01757027978473 \end{aligned}\] (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^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520) 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^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520, 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^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520); 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^12 - 45*x^10 + 1050*x^8 - 13530*x^6 + 99685*x^4 - 328905*x^2 + 317520); 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

$A_5:C_4$ (as 12T124):

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 non-solvable group of order 240
The 14 conjugacy class representatives for $A_5:C_4$
Character table for $A_5:C_4$

Intermediate fields

6.2.4706682753125.1

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 20 sibling: data not computed
Degree 24 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 12.4.110764312692821648486328125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{3}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(5\) Copy content Toggle raw display 5.1.12.11a1.4$x^{12} + 20$$12$$1$$11$$S_3 \times C_4$$$[\ ]_{12}^{2}$$
\(197\) Copy content Toggle raw display $\Q_{197}$$x + 195$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{197}$$x + 195$$1$$1$$0$Trivial$$[\ ]$$
197.2.5.8a1.1$x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$

Spectrum of ring of integers

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