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Properties

Label 12.0.133...625.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.339\times 10^{19}$
Root discriminant \(39.25\)
Ramified primes $5,17$
Class number $4$
Class group [4]
Galois group $A_5:C_4$ (as 12T124)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33)
 
gp: K = bnfinit(y^12 - 6*y^11 + 25*y^10 - 70*y^9 + 151*y^8 - 250*y^7 + 347*y^6 - 394*y^5 + 393*y^4 - 314*y^3 + 212*y^2 - 95*y + 33, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33)
 

\( x^{12} - 6 x^{11} + 25 x^{10} - 70 x^{9} + 151 x^{8} - 250 x^{7} + 347 x^{6} - 394 x^{5} + 393 x^{4} + \cdots + 33 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(13387459495169140625\) \(\medspace = 5^{8}\cdot 17^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(39.25\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{2/3}17^{11/12}\approx 39.254686577190164$
Ramified primes:   \(5\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{183}a^{11}+\frac{25}{183}a^{10}+\frac{7}{183}a^{9}+\frac{25}{183}a^{8}-\frac{50}{183}a^{7}-\frac{31}{183}a^{6}-\frac{4}{183}a^{5}+\frac{31}{183}a^{4}+\frac{4}{61}a^{3}+\frac{58}{183}a^{2}+\frac{58}{183}a-\frac{22}{61}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}$, which has order $4$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{20}{61}a^{11}+\frac{208}{183}a^{10}-\frac{725}{183}a^{9}+\frac{415}{61}a^{8}-\frac{586}{61}a^{7}+\frac{376}{61}a^{6}-\frac{347}{61}a^{5}-\frac{1006}{183}a^{4}+\frac{683}{183}a^{3}-\frac{2809}{183}a^{2}+\frac{1156}{183}a-\frac{449}{61}$, $\frac{20}{61}a^{11}-\frac{452}{183}a^{10}+\frac{1945}{183}a^{9}-\frac{1940}{61}a^{8}+\frac{4246}{61}a^{7}-\frac{7086}{61}a^{6}+\frac{9375}{61}a^{5}-\frac{31141}{183}a^{4}+\frac{28841}{183}a^{3}-\frac{22384}{183}a^{2}+\frac{12325}{183}a-\frac{1442}{61}$, $\frac{182}{183}a^{11}+\frac{940}{183}a^{10}-\frac{3653}{183}a^{9}+\frac{9175}{183}a^{8}-\frac{17801}{183}a^{7}+\frac{25772}{183}a^{6}-\frac{32029}{183}a^{5}+\frac{30958}{183}a^{4}-\frac{9085}{61}a^{3}+\frac{17077}{183}a^{2}-\frac{8726}{183}a+\frac{832}{61}$, $\frac{20}{183}a^{11}-\frac{110}{183}a^{10}+\frac{445}{183}a^{9}-\frac{1208}{183}a^{8}+\frac{846}{61}a^{7}-\frac{4097}{183}a^{6}+\frac{5471}{183}a^{5}-\frac{5480}{183}a^{4}+\frac{1422}{61}a^{3}-\frac{2195}{183}a^{2}+\frac{794}{183}a-\frac{13}{61}$, $\frac{16}{61}a^{11}-\frac{88}{61}a^{10}+\frac{356}{61}a^{9}-\frac{2765}{183}a^{8}+\frac{5591}{183}a^{7}-\frac{2753}{61}a^{6}+\frac{10361}{183}a^{5}-\frac{10285}{183}a^{4}+\frac{9238}{183}a^{3}-\frac{6061}{183}a^{2}+\frac{3089}{183}a-\frac{141}{61}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 27801.4421457893 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 27801.4421457893 \cdot 4}{2\cdot\sqrt{13387459495169140625}}\cr\approx \mathstrut & 0.935034091417998 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 151*x^8 - 250*x^7 + 347*x^6 - 394*x^5 + 393*x^4 - 314*x^3 + 212*x^2 - 95*x + 33);
 
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):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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

6.2.887410625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
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 equvalently 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 ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 5.12.8.1$x^{12} + 12 x^{10} + 32 x^{9} + 54 x^{8} + 96 x^{7} - 50 x^{6} + 240 x^{5} - 360 x^{4} - 884 x^{3} + 4044 x^{2} - 3912 x + 4173$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
\(17\) Copy content Toggle raw display 17.12.11.2$x^{12} + 34$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$