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Properties

Label 12.0.108...728.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.087\times 10^{24}$
Root discriminant \(100.70\)
Ramified primes $2,13,23$
Class number $36128$ (GRH)
Class group [2, 4, 4516] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557)
 
gp: K = bnfinit(y^12 + 299*y^10 + 34385*y^8 + 1898052*y^6 + 50931062*y^4 + 585707213*y^2 + 1924466557, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557)
 

\( x^{12} + 299x^{10} + 34385x^{8} + 1898052x^{6} + 50931062x^{4} + 585707213x^{2} + 1924466557 \) 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:   \(1086685418134968704585728\) \(\medspace = 2^{12}\cdot 13^{11}\cdot 23^{6}\) 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:  \(100.70\)
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:  $2\cdot 13^{11/12}23^{1/2}\approx 100.69517320952251$
Ramified primes:   \(2\), \(13\), \(23\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1196=2^{2}\cdot 13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1196}(1,·)$, $\chi_{1196}(643,·)$, $\chi_{1196}(737,·)$, $\chi_{1196}(551,·)$, $\chi_{1196}(1103,·)$, $\chi_{1196}(275,·)$, $\chi_{1196}(277,·)$, $\chi_{1196}(185,·)$, $\chi_{1196}(827,·)$, $\chi_{1196}(829,·)$, $\chi_{1196}(735,·)$, $\chi_{1196}(1013,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.18595408.2$^{2}$, 12.0.1086685418134968704585728.1$^{30}$

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

$1$, $a$, $\frac{1}{23}a^{2}$, $\frac{1}{23}a^{3}$, $\frac{1}{529}a^{4}$, $\frac{1}{529}a^{5}$, $\frac{1}{12167}a^{6}$, $\frac{1}{12167}a^{7}$, $\frac{1}{279841}a^{8}$, $\frac{1}{279841}a^{9}$, $\frac{1}{6436343}a^{10}$, $\frac{1}{6436343}a^{11}$ 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

$C_{2}\times C_{4}\times C_{4516}$, which has order $36128$ (assuming GRH)

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

Relative class number: $36128$ (assuming GRH)

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:   $\frac{1}{279841}a^{8}+\frac{8}{12167}a^{6}+\frac{20}{529}a^{4}+\frac{16}{23}a^{2}+2$, $\frac{1}{279841}a^{8}+\frac{9}{12167}a^{6}+\frac{27}{529}a^{4}+\frac{29}{23}a^{2}+6$, $\frac{1}{529}a^{4}+\frac{4}{23}a^{2}+2$, $\frac{1}{12167}a^{6}+\frac{6}{529}a^{4}+\frac{10}{23}a^{2}+4$, $\frac{1}{23}a^{2}+3$ 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:  \( 120.784031363 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 120.784031363 \cdot 36128}{2\cdot\sqrt{1086685418134968704585728}}\cr\approx \mathstrut & 0.128780686475 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557)
 
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 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557, 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 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557);
 
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 + 299*x^10 + 34385*x^8 + 1898052*x^6 + 50931062*x^4 + 585707213*x^2 + 1924466557);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.18595408.2, \(\Q(\zeta_{13})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }$ R ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.12.0.1}{12} }$

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 2.12.12.25$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$$2$$6$$12$$C_{12}$$[2]^{6}$
\(13\) Copy content Toggle raw display 13.12.11.4$x^{12} + 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
\(23\) Copy content Toggle raw display 23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$