Properties

Label 12.12.52206766144000000.1
Degree $12$
Signature $[12, 0]$
Discriminant $5.221\times 10^{16}$
Root discriminant \(24.73\)
Ramified primes $2,5,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4 \times C_2$ (as 12T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1)
 
gp: K = bnfinit(y^12 - 15*y^10 + 76*y^8 - 148*y^6 + 99*y^4 - 19*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1)
 

\( x^{12} - 15x^{10} + 76x^{8} - 148x^{6} + 99x^{4} - 19x^{2} + 1 \) 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:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(52206766144000000\) \(\medspace = 2^{12}\cdot 5^{6}\cdot 13^{8}\) 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:  \(24.73\)
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:  $2^{3/2}5^{1/2}13^{2/3}\approx 34.96704216279658$
Ramified primes:   \(2\), \(5\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a$ 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

Trivial group, which has order $1$ (assuming GRH)

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:  $11$
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:   $\frac{3}{5}a^{10}-\frac{44}{5}a^{8}+43a^{6}-\frac{389}{5}a^{4}+\frac{214}{5}a^{2}-\frac{19}{5}$, $\frac{4}{5}a^{11}-12a^{9}+\frac{303}{5}a^{7}-\frac{581}{5}a^{5}+\frac{358}{5}a^{3}-\frac{34}{5}a$, $a$, $\frac{3}{5}a^{10}-9a^{8}+\frac{226}{5}a^{6}-\frac{427}{5}a^{4}+\frac{261}{5}a^{2}-\frac{33}{5}$, $\frac{7}{5}a^{11}+\frac{1}{5}a^{10}-\frac{104}{5}a^{9}-3a^{8}+\frac{518}{5}a^{7}+\frac{77}{5}a^{6}-194a^{5}-\frac{154}{5}a^{4}+\frac{572}{5}a^{3}+\frac{97}{5}a^{2}-\frac{53}{5}a-\frac{6}{5}$, $\frac{12}{5}a^{11}-\frac{3}{5}a^{10}-\frac{178}{5}a^{9}+\frac{44}{5}a^{8}+\frac{882}{5}a^{7}-43a^{6}-\frac{1627}{5}a^{5}+\frac{389}{5}a^{4}+184a^{3}-\frac{214}{5}a^{2}-\frac{94}{5}a+\frac{24}{5}$, $\frac{3}{5}a^{11}+\frac{1}{5}a^{10}-\frac{44}{5}a^{9}-3a^{8}+43a^{7}+\frac{77}{5}a^{6}-\frac{389}{5}a^{5}-\frac{154}{5}a^{4}+\frac{214}{5}a^{3}+\frac{102}{5}a^{2}-\frac{19}{5}a-\frac{16}{5}$, $\frac{16}{5}a^{11}+\frac{4}{5}a^{10}-\frac{239}{5}a^{9}-12a^{8}+\frac{1201}{5}a^{7}+\frac{303}{5}a^{6}-\frac{2291}{5}a^{5}-\frac{581}{5}a^{4}+286a^{3}+\frac{363}{5}a^{2}-\frac{202}{5}a-\frac{54}{5}$, $\frac{9}{5}a^{11}-\frac{2}{5}a^{10}-\frac{134}{5}a^{9}+\frac{31}{5}a^{8}+\frac{667}{5}a^{7}-32a^{6}-\frac{1238}{5}a^{5}+\frac{306}{5}a^{4}+\frac{706}{5}a^{3}-\frac{181}{5}a^{2}-13a+\frac{16}{5}$, $\frac{9}{5}a^{11}+\frac{2}{5}a^{10}-\frac{134}{5}a^{9}-\frac{31}{5}a^{8}+\frac{667}{5}a^{7}+32a^{6}-\frac{1238}{5}a^{5}-\frac{306}{5}a^{4}+\frac{706}{5}a^{3}+\frac{181}{5}a^{2}-13a-\frac{16}{5}$, $\frac{4}{5}a^{11}-\frac{3}{5}a^{10}-12a^{9}+9a^{8}+\frac{303}{5}a^{7}-\frac{226}{5}a^{6}-\frac{581}{5}a^{5}+\frac{422}{5}a^{4}+\frac{363}{5}a^{3}-\frac{231}{5}a^{2}-\frac{59}{5}a+\frac{23}{5}$ Copy content Toggle raw display (assuming GRH)
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:  \( 27689.0510913 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 27689.0510913 \cdot 1}{2\cdot\sqrt{52206766144000000}}\cr\approx \mathstrut & 0.248184485115 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1)
 
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 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1, 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 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1);
 
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 - 15*x^10 + 76*x^8 - 148*x^6 + 99*x^4 - 19*x^2 + 1);
 
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

$C_2\times A_4$ (as 12T7):

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 solvable group of order 24
The 8 conjugacy class representatives for $A_4 \times C_2$
Character table for $A_4 \times C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.169.1, 6.6.9139520.1, 6.6.3570125.1, 6.6.45697600.1

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

Galois closure: deg 24
Degree 6 sibling: 6.6.9139520.1
Degree 8 sibling: 8.8.73116160000.1
Degree 12 sibling: deg 12
Minimal sibling: 6.6.9139520.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(2\) Copy content Toggle raw display 2.12.12.11$x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
\(5\) Copy content Toggle raw display 5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(13\) Copy content Toggle raw display 13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$