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Properties

Label 12.2.280755374552129536.1
Degree $12$
Signature $[2, 5]$
Discriminant $-2.808\times 10^{17}$
Root discriminant \(28.45\)
Ramified primes $2,17$
Class number $2$
Class group [2]
Galois group $S_4^2:D_4$ (as 12T260)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18)
 
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 + 4*y^9 + 2*y^8 - 28*y^7 + 28*y^6 + 11*y^5 - 15*y^4 + 64*y^3 - 100*y^2 + 54*y + 18, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18)
 

\( x^{12} - 3 x^{11} + 2 x^{10} + 4 x^{9} + 2 x^{8} - 28 x^{7} + 28 x^{6} + 11 x^{5} - 15 x^{4} + 64 x^{3} + \cdots + 18 \) 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:  $[2, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-280755374552129536\) \(\medspace = -\,2^{13}\cdot 17^{11}\) 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:  \(28.45\)
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:  $2^{3}17^{11/12}\approx 107.39931174612445$
Ramified primes:   \(2\), \(17\) 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:  \(\Q(\sqrt{-34}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{4266573207}a^{11}-\frac{286872280}{1422191069}a^{10}-\frac{1037465053}{4266573207}a^{9}+\frac{986459818}{4266573207}a^{8}-\frac{1291970911}{4266573207}a^{7}-\frac{187505560}{4266573207}a^{6}-\frac{798943634}{4266573207}a^{5}-\frac{2102552308}{4266573207}a^{4}-\frac{37826175}{1422191069}a^{3}+\frac{1337405635}{4266573207}a^{2}+\frac{798289586}{4266573207}a+\frac{695644082}{1422191069}$ 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

$C_{2}$, which has order $2$

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:  $6$
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:   $\frac{12615601}{1422191069}a^{11}-\frac{50780525}{1422191069}a^{10}+\frac{51840453}{1422191069}a^{9}+\frac{70510948}{1422191069}a^{8}-\frac{58711529}{1422191069}a^{7}-\frac{479950585}{1422191069}a^{6}+\frac{751624175}{1422191069}a^{5}+\frac{368998160}{1422191069}a^{4}-\frac{930547090}{1422191069}a^{3}+\frac{741701583}{1422191069}a^{2}-\frac{407644685}{1422191069}a+\frac{533544149}{1422191069}$, $\frac{110261017}{4266573207}a^{11}-\frac{84711694}{1422191069}a^{10}-\frac{11647294}{4266573207}a^{9}+\frac{536419795}{4266573207}a^{8}+\frac{683883071}{4266573207}a^{7}-\frac{2883301792}{4266573207}a^{6}+\frac{452294680}{4266573207}a^{5}+\frac{2777064065}{4266573207}a^{4}+\frac{559006150}{1422191069}a^{3}+\frac{4462154461}{4266573207}a^{2}-\frac{7096169989}{4266573207}a+\frac{302560857}{1422191069}$, $\frac{65854454}{1422191069}a^{11}-\frac{248900160}{1422191069}a^{10}+\frac{264450274}{1422191069}a^{9}+\frac{167100822}{1422191069}a^{8}-\frac{79659883}{1422191069}a^{7}-\frac{1872546570}{1422191069}a^{6}+\frac{3047350165}{1422191069}a^{5}-\frac{970620193}{1422191069}a^{4}-\frac{1202757984}{1422191069}a^{3}+\frac{6607378136}{1422191069}a^{2}-\frac{11075710637}{1422191069}a+\frac{4050078917}{1422191069}$, $\frac{19970948}{1422191069}a^{11}-\frac{71872799}{1422191069}a^{10}+\frac{97433807}{1422191069}a^{9}+\frac{26958628}{1422191069}a^{8}-\frac{6994685}{1422191069}a^{7}-\frac{571208224}{1422191069}a^{6}+\frac{809131867}{1422191069}a^{5}-\frac{470885093}{1422191069}a^{4}+\frac{546279421}{1422191069}a^{3}+\frac{2393460347}{1422191069}a^{2}-\frac{4209333813}{1422191069}a+\frac{2102765053}{1422191069}$, $\frac{174265933}{4266573207}a^{11}-\frac{73356811}{1422191069}a^{10}-\frac{275824858}{4266573207}a^{9}+\frac{574490977}{4266573207}a^{8}+\frac{1531171334}{4266573207}a^{7}-\frac{3123019525}{4266573207}a^{6}-\frac{2009544305}{4266573207}a^{5}+\frac{2536041323}{4266573207}a^{4}+\frac{683077724}{1422191069}a^{3}+\frac{12742103080}{4266573207}a^{2}+\frac{5227822526}{4266573207}a+\frac{297545863}{1422191069}$, $\frac{4523141305}{4266573207}a^{11}-\frac{4846155070}{1422191069}a^{10}+\frac{12303871469}{4266573207}a^{9}+\frac{15306608554}{4266573207}a^{8}+\frac{5226755789}{4266573207}a^{7}-\frac{126166351861}{4266573207}a^{6}+\frac{152964669256}{4266573207}a^{5}+\frac{14604079250}{4266573207}a^{4}-\frac{23340298324}{1422191069}a^{3}+\frac{301236197116}{4266573207}a^{2}-\frac{514883948038}{4266573207}a+\frac{118756478537}{1422191069}$ 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:  \( 23296.9609907 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 23296.9609907 \cdot 2}{2\cdot\sqrt{280755374552129536}}\cr\approx \mathstrut & 1.72224354707 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18)
 
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 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18, 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 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18);
 
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 + 2*x^10 + 4*x^9 + 2*x^8 - 28*x^7 + 28*x^6 + 11*x^5 - 15*x^4 + 64*x^3 - 100*x^2 + 54*x + 18);
 
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

$S_4^2:D_4$ (as 12T260):

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 4608
The 65 conjugacy class representatives for $S_4^2:D_4$
Character table for $S_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 6.2.11358856.1

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

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 siblings: data not computed
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 R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ R ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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]])
 
// 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 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
\(17\) Copy content Toggle raw display 17.12.11.2$x^{12} + 34$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$