# Properties

 Label 12.0.126759838761.1 Degree $12$ Signature $[0, 6]$ Discriminant $126759838761$ Root discriminant $$8.42$$ Ramified primes $3,43$ Class number $1$ Class group trivial Galois group $\SOPlus(4,2)$ (as 12T35)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)

gp: K = bnfinit(y^12 - 3*y^11 + 6*y^10 - 11*y^9 + 15*y^8 - 18*y^7 + 21*y^6 - 18*y^5 + 15*y^4 - 11*y^3 + 6*y^2 - 3*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*x^2 - 3*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*x^2 - 3*x + 1)

$$x^{12} - 3 x^{11} + 6 x^{10} - 11 x^{9} + 15 x^{8} - 18 x^{7} + 21 x^{6} - 18 x^{5} + 15 x^{4} - 11 x^{3} + 6 x^{2} - 3 x + 1$$

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: $$126759838761$$ 126759838761 $$\medspace = 3^{13}\cdot 43^{3}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$8.42$$ 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: $$3$$, $$43$$ 3, 43 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{129})$$ $\card{ \Aut(K/\Q) }$: $6$ 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 $$a$$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: Yes Index: $1$ Inessential primes: None

## 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: $$a^{11} - 2 a^{10} + 3 a^{9} - 5 a^{8} + 5 a^{7} - 6 a^{6} + 8 a^{5} - 5 a^{4} + 6 a^{3} - 4 a^{2} + 3 a - 1$$ a^(11) - 2*a^(10) + 3*a^(9) - 5*a^(8) + 5*a^(7) - 6*a^(6) + 8*a^(5) - 5*a^(4) + 6*a^(3) - 4*a^(2) + 3*a - 1  (order $6$) 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: $2a^{11}-5a^{10}+10a^{9}-19a^{8}+25a^{7}-31a^{6}+36a^{5}-28a^{4}+25a^{3}-16a^{2}+8a-3$, $a^{11}-3a^{10}+7a^{9}-13a^{8}+18a^{7}-23a^{6}+25a^{5}-21a^{4}+19a^{3}-10a^{2}+6a-3$, $a^{11}-2a^{10}+2a^{9}-2a^{8}-a^{7}+5a^{6}-6a^{5}+10a^{4}-10a^{3}+7a^{2}-4a+2$, $a^{11}-3a^{10}+6a^{9}-11a^{8}+15a^{7}-18a^{6}+21a^{5}-18a^{4}+15a^{3}-11a^{2}+5a-3$, $a^{11}-3a^{10}+7a^{9}-13a^{8}+18a^{7}-22a^{6}+23a^{5}-19a^{4}+16a^{3}-8a^{2}+5a-1$ 2*a^11 - 5*a^10 + 10*a^9 - 19*a^8 + 25*a^7 - 31*a^6 + 36*a^5 - 28*a^4 + 25*a^3 - 16*a^2 + 8*a - 3, a^11 - 3*a^10 + 7*a^9 - 13*a^8 + 18*a^7 - 23*a^6 + 25*a^5 - 21*a^4 + 19*a^3 - 10*a^2 + 6*a - 3, a^11 - 2*a^10 + 2*a^9 - 2*a^8 - a^7 + 5*a^6 - 6*a^5 + 10*a^4 - 10*a^3 + 7*a^2 - 4*a + 2, a^11 - 3*a^10 + 6*a^9 - 11*a^8 + 15*a^7 - 18*a^6 + 21*a^5 - 18*a^4 + 15*a^3 - 11*a^2 + 5*a - 3, a^11 - 3*a^10 + 7*a^9 - 13*a^8 + 18*a^7 - 22*a^6 + 23*a^5 - 19*a^4 + 16*a^3 - 8*a^2 + 5*a - 1 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: $$8.68701246184$$ 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 8.68701246184 \cdot 1}{6\cdot\sqrt{126759838761}}\cr\approx \mathstrut & 0.250211672130 \end{aligned}

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

x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*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 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*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 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*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 + 6*x^10 - 11*x^9 + 15*x^8 - 18*x^7 + 21*x^6 - 18*x^5 + 15*x^4 - 11*x^3 + 6*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

$\SOPlus(4,2)$ (as 12T35):

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 72 The 9 conjugacy class representatives for $\SOPlus(4,2)$ Character table for $\SOPlus(4,2)$

## 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: 6.4.57960603.1, 6.0.31347.1 Degree 9 sibling: 9.3.4694808843.1 Degree 12 siblings: deg 12, deg 12, deg 12, deg 12, deg 12 Degree 18 siblings: 18.0.66123690216932995947.1, deg 18, deg 18 Degree 24 siblings: data not computed Degree 36 siblings: data not computed Minimal sibling: 6.0.31347.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.4.0.1}{4} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ R ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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]])