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Properties

Label 12.12.4864965285308625.1
Degree $12$
Signature $[12, 0]$
Discriminant $4.865\times 10^{15}$
Root discriminant \(20.29\)
Ramified primes $3,5,7$
Class number $1$
Class group trivial
Galois group $D_4 \times C_3$ (as 12T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*x + 1)
 
gp: K = bnfinit(y^12 - 4*y^11 - 9*y^10 + 45*y^9 + 10*y^8 - 150*y^7 + 56*y^6 + 135*y^5 - 36*y^4 - 53*y^3 - y^2 + 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*x + 1)
 

\( x^{12} - 4 x^{11} - 9 x^{10} + 45 x^{9} + 10 x^{8} - 150 x^{7} + 56 x^{6} + 135 x^{5} - 36 x^{4} + \cdots + 1 \) 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:   \(4864965285308625\) \(\medspace = 3^{9}\cdot 5^{3}\cdot 7^{11}\) 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:  \(20.29\)
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:  $3^{3/4}5^{1/2}7^{11/12}\approx 30.33885222204471$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{105}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{43}a^{11}+\frac{16}{43}a^{10}+\frac{10}{43}a^{9}-\frac{13}{43}a^{8}+\frac{8}{43}a^{7}+\frac{10}{43}a^{6}-\frac{2}{43}a^{5}+\frac{9}{43}a^{4}+\frac{15}{43}a^{3}-\frac{11}{43}a^{2}-\frac{6}{43}a+\frac{15}{43}$ 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 $1$

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:   $\frac{902}{43}a^{11}-\frac{3929}{43}a^{10}-\frac{6718}{43}a^{9}+\frac{42970}{43}a^{8}-\frac{6286}{43}a^{7}-\frac{132923}{43}a^{6}+\frac{97827}{43}a^{5}+\frac{86421}{43}a^{4}-\frac{63139}{43}a^{3}-\frac{24757}{43}a^{2}+\frac{7918}{43}a+\frac{2479}{43}$, $\frac{1146}{43}a^{11}-\frac{5013}{43}a^{10}-\frac{8449}{43}a^{9}+\frac{54762}{43}a^{8}-\frac{8892}{43}a^{7}-\frac{168882}{43}a^{6}+\frac{126837}{43}a^{5}+\frac{108225}{43}a^{4}-\frac{81108}{43}a^{3}-\frac{31182}{43}a^{2}+\frac{10152}{43}a+\frac{3172}{43}$, $\frac{1158}{43}a^{11}-\frac{5079}{43}a^{10}-\frac{8458}{43}a^{9}+\frac{55380}{43}a^{8}-\frac{9871}{43}a^{7}-\frac{169966}{43}a^{6}+\frac{130898}{43}a^{5}+\frac{106054}{43}a^{4}-\frac{83551}{43}a^{3}-\frac{29250}{43}a^{2}+\frac{10553}{43}a+\frac{2922}{43}$, $\frac{451}{43}a^{11}-\frac{1986}{43}a^{10}-\frac{3273}{43}a^{9}+\frac{21657}{43}a^{8}-\frac{4046}{43}a^{7}-\frac{66440}{43}a^{6}+\frac{51472}{43}a^{5}+\frac{41340}{43}a^{4}-\frac{32494}{43}a^{3}-\frac{11411}{43}a^{2}+\frac{3873}{43}a+\frac{1089}{43}$, $\frac{175}{43}a^{11}-\frac{726}{43}a^{10}-\frac{1475}{43}a^{9}+\frac{8131}{43}a^{8}+\frac{583}{43}a^{7}-\frac{26716}{43}a^{6}+\frac{14012}{43}a^{5}+\frac{22559}{43}a^{4}-\frac{11006}{43}a^{3}-\frac{7816}{43}a^{2}+\frac{1616}{43}a+\frac{819}{43}$, $\frac{1201}{43}a^{11}-\frac{5251}{43}a^{10}-\frac{8845}{43}a^{9}+\frac{57315}{43}a^{8}-\frac{9441}{43}a^{7}-\frac{176416}{43}a^{6}+\frac{133306}{43}a^{5}+\frac{111859}{43}a^{4}-\frac{85099}{43}a^{3}-\frac{31529}{43}a^{2}+\frac{10510}{43}a+\frac{3180}{43}$, $\frac{750}{43}a^{11}-\frac{3265}{43}a^{10}-\frac{5572}{43}a^{9}+\frac{35658}{43}a^{8}-\frac{5395}{43}a^{7}-\frac{109976}{43}a^{6}+\frac{81834}{43}a^{5}+\frac{70519}{43}a^{4}-\frac{52605}{43}a^{3}-\frac{20118}{43}a^{2}+\frac{6637}{43}a+\frac{2048}{43}$, $\frac{175}{43}a^{11}-\frac{726}{43}a^{10}-\frac{1475}{43}a^{9}+\frac{8131}{43}a^{8}+\frac{583}{43}a^{7}-\frac{26716}{43}a^{6}+\frac{14012}{43}a^{5}+\frac{22559}{43}a^{4}-\frac{11006}{43}a^{3}-\frac{7816}{43}a^{2}+\frac{1573}{43}a+\frac{819}{43}$, $\frac{200}{43}a^{11}-\frac{928}{43}a^{10}-\frac{1268}{43}a^{9}+\frac{9999}{43}a^{8}-\frac{3732}{43}a^{7}-\frac{29562}{43}a^{6}+\frac{28453}{43}a^{5}+\frac{14571}{43}a^{4}-\frac{16995}{43}a^{3}-\frac{3060}{43}a^{2}+\frac{2025}{43}a+\frac{291}{43}$, $\frac{493}{43}a^{11}-\frac{2174}{43}a^{10}-\frac{3584}{43}a^{9}+\frac{23777}{43}a^{8}-\frac{4398}{43}a^{7}-\frac{73502}{43}a^{6}+\frac{56548}{43}a^{5}+\frac{47738}{43}a^{4}-\frac{36895}{43}a^{3}-\frac{14410}{43}a^{2}+\frac{4868}{43}a+\frac{1547}{43}$, $a^{11}-4a^{10}-9a^{9}+45a^{8}+10a^{7}-150a^{6}+56a^{5}+135a^{4}-36a^{3}-53a^{2}+5$ 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:  \( 6092.78082056 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 6092.78082056 \cdot 1}{2\cdot\sqrt{4864965285308625}}\cr\approx \mathstrut & 0.178898061527 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*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 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*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 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*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 - 4*x^11 - 9*x^10 + 45*x^9 + 10*x^8 - 150*x^7 + 56*x^6 + 135*x^5 - 36*x^4 - 53*x^3 - x^2 + 6*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

$C_3\times D_4$ (as 12T14):

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 15 conjugacy class representatives for $D_4 \times C_3$
Character table for $D_4 \times C_3$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), 4.4.46305.1, \(\Q(\zeta_{21})^+\)

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 12 sibling: deg 12
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 ${\href{/padicField/2.6.0.1}{6} }^{2}$ R R R ${\href{/padicField/11.12.0.1}{12} }$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(3\) Copy content Toggle raw display 3.12.9.1$x^{12} + 18 x^{4} - 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
\(5\) Copy content Toggle raw display 5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.12.11.2$x^{12} + 7$$12$$1$$11$$D_4 \times C_3$$[\ ]_{12}^{2}$