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Properties

Label 12.4.986...625.1
Degree $12$
Signature $[4, 4]$
Discriminant $9.869\times 10^{25}$
Root discriminant \(146.62\)
Ramified primes $3,5,41$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times A_5$ (as 12T76)

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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 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743)
 
Copy content gp:K = bnfinit(y^12 - y^11 + 22*y^10 + 65*y^9 - 755*y^8 + 3669*y^7 - 13894*y^6 + 8118*y^5 + 65920*y^4 - 295700*y^3 - 688589*y^2 - 854921*y - 753743, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743)
 

\( x^{12} - x^{11} + 22 x^{10} + 65 x^{9} - 755 x^{8} + 3669 x^{7} - 13894 x^{6} + 8118 x^{5} + \cdots - 753743 \) 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:   \(98690634698303375244140625\) \(\medspace = 3^{4}\cdot 5^{16}\cdot 41^{8}\) 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:  \(146.62\)
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:  $3^{1/2}5^{8/5}41^{4/5}\approx 443.7513830832364$
Ramified primes:   \(3\), \(5\), \(41\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{41}a^{7}+\frac{18}{41}a^{6}-\frac{10}{41}a^{5}+\frac{9}{41}a^{4}-\frac{19}{41}a^{3}-\frac{1}{41}a^{2}+\frac{11}{41}a-\frac{14}{41}$, $\frac{1}{41}a^{8}-\frac{6}{41}a^{6}-\frac{16}{41}a^{5}-\frac{17}{41}a^{4}+\frac{13}{41}a^{3}-\frac{12}{41}a^{2}-\frac{7}{41}a+\frac{6}{41}$, $\frac{1}{41}a^{9}+\frac{10}{41}a^{6}+\frac{5}{41}a^{5}-\frac{15}{41}a^{4}-\frac{3}{41}a^{3}-\frac{13}{41}a^{2}-\frac{10}{41}a-\frac{2}{41}$, $\frac{1}{123}a^{10}+\frac{1}{123}a^{9}+\frac{1}{123}a^{8}-\frac{1}{123}a^{7}+\frac{19}{41}a^{6}+\frac{2}{123}a^{5}+\frac{10}{41}a^{4}-\frac{40}{123}a^{3}+\frac{17}{123}a^{2}-\frac{17}{123}a+\frac{35}{123}$, $\frac{1}{29\cdots 88}a^{11}+\frac{93\cdots 96}{73\cdots 97}a^{10}-\frac{15\cdots 55}{14\cdots 94}a^{9}+\frac{47\cdots 05}{98\cdots 96}a^{8}-\frac{67\cdots 09}{73\cdots 97}a^{7}-\frac{53\cdots 87}{29\cdots 88}a^{6}+\frac{89\cdots 67}{29\cdots 88}a^{5}-\frac{17\cdots 71}{29\cdots 88}a^{4}+\frac{85\cdots 45}{29\cdots 88}a^{3}-\frac{39\cdots 13}{98\cdots 96}a^{2}+\frac{74\cdots 61}{24\cdots 99}a-\frac{30\cdots 25}{29\cdots 88}$ 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:  $3$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (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:  Trivial group, which has order $1$ (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:   $\frac{19\cdots 51}{29\cdots 88}a^{11}-\frac{11\cdots 96}{73\cdots 97}a^{10}+\frac{25\cdots 05}{14\cdots 94}a^{9}+\frac{56\cdots 29}{29\cdots 88}a^{8}-\frac{12\cdots 35}{24\cdots 99}a^{7}+\frac{96\cdots 31}{29\cdots 88}a^{6}-\frac{14\cdots 17}{98\cdots 96}a^{5}+\frac{79\cdots 15}{29\cdots 88}a^{4}-\frac{72\cdots 97}{29\cdots 88}a^{3}-\frac{61\cdots 89}{29\cdots 88}a^{2}-\frac{11\cdots 71}{73\cdots 97}a-\frac{33\cdots 89}{98\cdots 96}$, $\frac{43\cdots 31}{49\cdots 98}a^{11}-\frac{76\cdots 36}{24\cdots 99}a^{10}+\frac{61\cdots 11}{24\cdots 99}a^{9}+\frac{16\cdots 73}{49\cdots 98}a^{8}-\frac{18\cdots 11}{24\cdots 99}a^{7}+\frac{24\cdots 21}{49\cdots 98}a^{6}-\frac{11\cdots 31}{49\cdots 98}a^{5}+\frac{26\cdots 99}{49\cdots 98}a^{4}-\frac{14\cdots 87}{49\cdots 98}a^{3}-\frac{14\cdots 41}{49\cdots 98}a^{2}+\frac{24\cdots 31}{24\cdots 99}a-\frac{43\cdots 05}{11\cdots 78}$, $\frac{26\cdots 43}{98\cdots 96}a^{11}-\frac{70\cdots 57}{73\cdots 97}a^{10}-\frac{28\cdots 75}{14\cdots 94}a^{9}-\frac{73\cdots 21}{29\cdots 88}a^{8}-\frac{35\cdots 26}{73\cdots 97}a^{7}+\frac{24\cdots 51}{98\cdots 96}a^{6}-\frac{86\cdots 77}{29\cdots 88}a^{5}+\frac{68\cdots 83}{98\cdots 96}a^{4}+\frac{30\cdots 13}{29\cdots 88}a^{3}+\frac{59\cdots 33}{29\cdots 88}a^{2}+\frac{16\cdots 75}{73\cdots 97}a+\frac{50\cdots 03}{29\cdots 88}$, $\frac{24\cdots 93}{29\cdots 88}a^{11}+\frac{58\cdots 34}{24\cdots 99}a^{10}-\frac{11\cdots 19}{49\cdots 98}a^{9}-\frac{34\cdots 19}{29\cdots 88}a^{8}+\frac{48\cdots 37}{73\cdots 97}a^{7}-\frac{12\cdots 61}{29\cdots 88}a^{6}+\frac{57\cdots 85}{29\cdots 88}a^{5}-\frac{12\cdots 85}{29\cdots 88}a^{4}+\frac{24\cdots 45}{98\cdots 96}a^{3}+\frac{59\cdots 87}{29\cdots 88}a^{2}+\frac{14\cdots 56}{73\cdots 97}a+\frac{24\cdots 17}{71\cdots 68}$, $\frac{27\cdots 49}{98\cdots 96}a^{11}+\frac{24\cdots 91}{73\cdots 97}a^{10}+\frac{13\cdots 37}{14\cdots 94}a^{9}+\frac{10\cdots 17}{29\cdots 88}a^{8}-\frac{80\cdots 75}{73\cdots 97}a^{7}+\frac{77\cdots 85}{98\cdots 96}a^{6}-\frac{13\cdots 31}{29\cdots 88}a^{5}-\frac{29\cdots 71}{98\cdots 96}a^{4}-\frac{11\cdots 57}{71\cdots 68}a^{3}-\frac{27\cdots 89}{29\cdots 88}a^{2}+\frac{30\cdots 06}{73\cdots 97}a+\frac{54\cdots 37}{29\cdots 88}$, $\frac{24\cdots 21}{98\cdots 96}a^{11}-\frac{52\cdots 57}{73\cdots 97}a^{10}+\frac{99\cdots 83}{14\cdots 94}a^{9}+\frac{10\cdots 77}{29\cdots 88}a^{8}-\frac{14\cdots 08}{73\cdots 97}a^{7}+\frac{12\cdots 61}{98\cdots 96}a^{6}-\frac{17\cdots 15}{29\cdots 88}a^{5}+\frac{30\cdots 97}{23\cdots 56}a^{4}-\frac{21\cdots 37}{29\cdots 88}a^{3}-\frac{17\cdots 61}{29\cdots 88}a^{2}-\frac{44\cdots 43}{73\cdots 97}a-\frac{29\cdots 39}{29\cdots 88}$, $\frac{30\cdots 57}{98\cdots 96}a^{11}+\frac{14\cdots 07}{73\cdots 97}a^{10}-\frac{87\cdots 37}{14\cdots 94}a^{9}-\frac{79\cdots 25}{29\cdots 88}a^{8}+\frac{16\cdots 52}{73\cdots 97}a^{7}-\frac{86\cdots 45}{98\cdots 96}a^{6}+\frac{81\cdots 71}{29\cdots 88}a^{5}+\frac{35\cdots 43}{98\cdots 96}a^{4}-\frac{88\cdots 91}{29\cdots 88}a^{3}+\frac{21\cdots 81}{29\cdots 88}a^{2}+\frac{26\cdots 32}{73\cdots 97}a+\frac{82\cdots 27}{29\cdots 88}$ 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:  \( 273555130.41668713 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 273555130.41668713 \cdot 1}{2\cdot\sqrt{98690634698303375244140625}}\cr\approx \mathstrut & 0.343333636622524 \end{aligned}\] (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 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743) 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 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743, 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 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743); 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 - x^11 + 22*x^10 + 65*x^9 - 755*x^8 + 3669*x^7 - 13894*x^6 + 8118*x^5 + 65920*x^4 - 295700*x^3 - 688589*x^2 - 854921*x - 753743); 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

$C_2\times A_5$ (as 12T76):

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 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

6.2.9934316015625.1

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 10 sibling: data not computed
Degree 12 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: 10.0.296071904094910125732421875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ R R ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ R ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(3\) Copy content Toggle raw display 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.5.16a4.5$x^{10} + 20 x^{9} + 190 x^{8} + 1120 x^{7} + 4360 x^{6} + 11104 x^{5} + 17840 x^{4} + 17280 x^{3} + 9680 x^{2} + 2880 x + 457$$5$$2$$16$$C_{10}$$$[2]^{2}$$
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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