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Properties

Label 12.0.101...568.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.016\times 10^{29}$
Root discriminant \(261.35\)
Ramified primes $2,3,7,13$
Class number $1814800$ (GRH)
Class group [2, 2, 10, 45370] (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 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592)
 
gp: K = bnfinit(y^12 + 1092*y^10 + 286650*y^8 + 29857464*y^6 + 1314691560*y^4 + 22086818208*y^2 + 115955795592, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592)
 

\( x^{12} + 1092x^{10} + 286650x^{8} + 29857464x^{6} + 1314691560x^{4} + 22086818208x^{2} + 115955795592 \) 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:   \(101563848359892128205077741568\) \(\medspace = 2^{33}\cdot 3^{6}\cdot 7^{6}\cdot 13^{10}\) 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:  \(261.35\)
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:  $2^{11/4}3^{1/2}7^{1/2}13^{5/6}\approx 261.35346503452513$
Ramified primes:   \(2\), \(3\), \(7\), \(13\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4368=2^{4}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4368}(3611,·)$, $\chi_{4368}(1,·)$, $\chi_{4368}(1091,·)$, $\chi_{4368}(4033,·)$, $\chi_{4368}(841,·)$, $\chi_{4368}(3275,·)$, $\chi_{4368}(2435,·)$, $\chi_{4368}(3025,·)$, $\chi_{4368}(1427,·)$, $\chi_{4368}(2185,·)$, $\chi_{4368}(1849,·)$, $\chi_{4368}(251,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.152635392.5$^{2}$, 12.0.101563848359892128205077741568.1$^{30}$

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

$1$, $a$, $\frac{1}{21}a^{2}$, $\frac{1}{21}a^{3}$, $\frac{1}{882}a^{4}$, $\frac{1}{882}a^{5}$, $\frac{1}{240786}a^{6}$, $\frac{1}{240786}a^{7}$, $\frac{1}{50565060}a^{8}+\frac{1}{2205}a^{4}+\frac{2}{5}$, $\frac{1}{50565060}a^{9}+\frac{1}{2205}a^{5}+\frac{2}{5}a$, $\frac{1}{32917854060}a^{10}-\frac{1}{261252810}a^{8}-\frac{59}{37321830}a^{6}-\frac{1}{3255}a^{4}+\frac{11}{465}a^{2}-\frac{22}{155}$, $\frac{1}{32917854060}a^{11}-\frac{1}{261252810}a^{9}-\frac{59}{37321830}a^{7}-\frac{1}{3255}a^{5}+\frac{11}{465}a^{3}-\frac{22}{155}a$ 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

$C_{2}\times C_{2}\times C_{10}\times C_{45370}$, which has order $1814800$ (assuming GRH)

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:   \( -1 \)  (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:   $\frac{1}{6583570812}a^{10}+\frac{2}{11196549}a^{8}+\frac{29}{533169}a^{6}+\frac{8}{1519}a^{4}+\frac{11}{93}a^{2}-\frac{53}{31}$, $\frac{1}{1266071310}a^{10}+\frac{127}{156751686}a^{8}+\frac{251}{1435455}a^{6}+\frac{383}{27342}a^{4}+\frac{1382}{3255}a^{2}+\frac{84}{31}$, $\frac{1}{783758430}a^{10}+\frac{61}{44786196}a^{8}+\frac{53}{159495}a^{6}+\frac{13}{434}a^{4}+\frac{154}{155}a^{2}+\frac{305}{31}$, $\frac{73}{32917854060}a^{10}+\frac{737}{313503372}a^{8}+\frac{499}{888615}a^{6}+\frac{673}{13671}a^{4}+\frac{1667}{1085}a^{2}+\frac{367}{31}$, $\frac{47}{32917854060}a^{10}+\frac{23}{14928732}a^{8}+\frac{7216}{18660915}a^{6}+\frac{107}{3038}a^{4}+\frac{517}{465}a^{2}+\frac{252}{31}$ 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:  \( 930.6875006453596 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 930.6875006453596 \cdot 1814800}{2\cdot\sqrt{101563848359892128205077741568}}\cr\approx \mathstrut & 0.163046804698340 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592)
 
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 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592, 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 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592);
 
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 + 1092*x^10 + 286650*x^8 + 29857464*x^6 + 1314691560*x^4 + 22086818208*x^2 + 115955795592);
 
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

\(\Q(\sqrt{2}) \), 3.3.169.1, 4.0.152635392.5, 6.6.14623232.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }$

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 2.12.33.344$x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
\(3\) Copy content Toggle raw display 3.12.6.1$x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
\(7\) Copy content Toggle raw display 7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(13\) Copy content Toggle raw display 13.12.10.5$x^{12} - 1586 x^{6} - 198575$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$