# Properties

 Label 12.0.470596000000.1 Degree $12$ Signature $[0, 6]$ Discriminant $470596000000$ Root discriminant $$9.39$$ Ramified primes $2,5,7$ Class number $1$ Class group trivial Galois group $D_6$ (as 12T3)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1)

gp: K = bnfinit(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1)

$$x^{12} - 3 x^{11} + 5 x^{10} - 8 x^{9} + 11 x^{8} - 13 x^{7} + 18 x^{6} - 13 x^{5} + 11 x^{4} - 8 x^{3} + 5 x^{2} - 3 x + 1$$

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: $$470596000000$$ 470596000000 $$\medspace = 2^{8}\cdot 5^{6}\cdot 7^{6}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$9.39$$ 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)) Ramified primes: $$2$$, $$5$$, $$7$$ 2, 5, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q$$ $\card{ \Gal(K/\Q) }$: $12$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{10}a^{10}-\frac{1}{10}a^{8}-\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{1}{10}a^{4}+\frac{2}{5}a^{3}-\frac{1}{10}a^{2}-\frac{1}{2}a+\frac{1}{10}$, $\frac{1}{20}a^{11}-\frac{1}{20}a^{9}-\frac{1}{20}a^{8}+\frac{1}{5}a^{7}-\frac{1}{4}a^{6}-\frac{1}{20}a^{5}+\frac{1}{5}a^{4}+\frac{9}{20}a^{3}+\frac{1}{4}a^{2}-\frac{1}{5}a+\frac{1}{4}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: No Index: Not computed Inessential primes: $2$

## Class group and class number

Trivial group, which has order $1$

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: $$-1$$ -1  (order $2$) 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: $\frac{7}{10}a^{11}-\frac{19}{10}a^{10}+\frac{33}{10}a^{9}-\frac{53}{10}a^{8}+\frac{67}{10}a^{7}-\frac{81}{10}a^{6}+\frac{113}{10}a^{5}-\frac{63}{10}a^{4}+\frac{77}{10}a^{3}-\frac{21}{10}a^{2}+\frac{27}{10}a-\frac{19}{10}$, $\frac{2}{5}a^{11}-\frac{3}{10}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{11}{10}a^{7}+\frac{9}{5}a^{6}-\frac{2}{5}a^{5}+\frac{59}{10}a^{4}-\frac{3}{5}a^{3}+\frac{9}{5}a^{2}-\frac{1}{10}a+\frac{1}{5}$, $\frac{13}{20}a^{11}-\frac{13}{10}a^{10}+\frac{37}{20}a^{9}-\frac{67}{20}a^{8}+\frac{39}{10}a^{7}-\frac{89}{20}a^{6}+\frac{147}{20}a^{5}-\frac{11}{10}a^{4}+\frac{103}{20}a^{3}-\frac{9}{20}a^{2}+\frac{9}{10}a-\frac{11}{20}$, $\frac{19}{20}a^{11}-\frac{23}{10}a^{10}+\frac{71}{20}a^{9}-\frac{123}{20}a^{8}+\frac{81}{10}a^{7}-\frac{189}{20}a^{6}+\frac{281}{20}a^{5}-\frac{69}{10}a^{4}+\frac{197}{20}a^{3}-\frac{109}{20}a^{2}+\frac{27}{10}a-\frac{31}{20}$, $\frac{17}{20}a^{11}-\frac{12}{5}a^{10}+\frac{73}{20}a^{9}-\frac{119}{20}a^{8}+\frac{83}{10}a^{7}-\frac{187}{20}a^{6}+\frac{263}{20}a^{5}-\frac{41}{5}a^{4}+\frac{131}{20}a^{3}-\frac{117}{20}a^{2}+\frac{21}{10}a-\frac{13}{20}$ 7/10*a^11 - 19/10*a^10 + 33/10*a^9 - 53/10*a^8 + 67/10*a^7 - 81/10*a^6 + 113/10*a^5 - 63/10*a^4 + 77/10*a^3 - 21/10*a^2 + 27/10*a - 19/10, 2/5*a^11 - 3/10*a^10 - 2/5*a^9 + 2/5*a^8 - 11/10*a^7 + 9/5*a^6 - 2/5*a^5 + 59/10*a^4 - 3/5*a^3 + 9/5*a^2 - 1/10*a + 1/5, 13/20*a^11 - 13/10*a^10 + 37/20*a^9 - 67/20*a^8 + 39/10*a^7 - 89/20*a^6 + 147/20*a^5 - 11/10*a^4 + 103/20*a^3 - 9/20*a^2 + 9/10*a - 11/20, 19/20*a^11 - 23/10*a^10 + 71/20*a^9 - 123/20*a^8 + 81/10*a^7 - 189/20*a^6 + 281/20*a^5 - 69/10*a^4 + 197/20*a^3 - 109/20*a^2 + 27/10*a - 31/20, 17/20*a^11 - 12/5*a^10 + 73/20*a^9 - 119/20*a^8 + 83/10*a^7 - 187/20*a^6 + 263/20*a^5 - 41/5*a^4 + 131/20*a^3 - 117/20*a^2 + 21/10*a - 13/20 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: $$6.67082664998$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 6.67082664998 \cdot 1}{2\cdot\sqrt{470596000000}}\approx 0.299160846811$

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 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 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 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 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 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 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 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

$D_6$ (as 12T3):

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 12 The 6 conjugacy class representatives for $D_6$ Character table for $D_6$

## Intermediate fields

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 6 siblings: 6.0.137200.1, 6.2.98000.1

## 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 R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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]])