Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 1)
gp: K = bnfinit(y^12 + 2*y^10 + 7*y^8 + 8*y^6 + 11*y^4 - 5*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 1)
\( x^{12} + 2x^{10} + 7x^{8} + 8x^{6} + 11x^{4} - 5x^{2} + 1 \)
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: |
\(47608675209216\)
\(\medspace = 2^{12}\cdot 3^{8}\cdot 11^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.80\) | 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: | $2\cdot 3^{4/3}11^{1/2}\approx 28.700404072565842$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{95}a^{10}-\frac{27}{95}a^{8}+\frac{6}{19}a^{6}-\frac{7}{95}a^{4}+\frac{24}{95}a^{2}-\frac{36}{95}$, $\frac{1}{95}a^{11}-\frac{27}{95}a^{9}+\frac{6}{19}a^{7}-\frac{7}{95}a^{5}+\frac{24}{95}a^{3}-\frac{36}{95}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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: |
\( -\frac{11}{19} a^{11} - \frac{26}{19} a^{9} - \frac{83}{19} a^{7} - \frac{113}{19} a^{5} - \frac{150}{19} a^{3} + \frac{16}{19} a \)
(order $4$)
| 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: |
$\frac{13}{95}a^{10}+\frac{29}{95}a^{8}+\frac{21}{19}a^{6}+\frac{99}{95}a^{4}+\frac{217}{95}a^{2}-\frac{88}{95}$, $\frac{26}{95}a^{11}+\frac{58}{95}a^{9}+\frac{42}{19}a^{7}+\frac{293}{95}a^{5}+\frac{434}{95}a^{3}+\frac{14}{95}a$, $\frac{68}{95}a^{11}+\frac{29}{95}a^{10}+\frac{159}{95}a^{9}+\frac{72}{95}a^{8}+\frac{104}{19}a^{7}+\frac{41}{19}a^{6}+\frac{664}{95}a^{5}+\frac{272}{95}a^{4}+\frac{872}{95}a^{3}+\frac{316}{95}a^{2}-\frac{168}{95}a+\frac{1}{95}$, $\frac{13}{95}a^{11}+\frac{33}{95}a^{10}+\frac{29}{95}a^{9}+\frac{59}{95}a^{8}+\frac{21}{19}a^{7}+\frac{46}{19}a^{6}+\frac{99}{95}a^{5}+\frac{244}{95}a^{4}+\frac{217}{95}a^{3}+\frac{317}{95}a^{2}-\frac{88}{95}a-\frac{48}{95}$, $\frac{149}{95}a^{11}-\frac{88}{95}a^{10}+\frac{347}{95}a^{9}-\frac{189}{95}a^{8}+\frac{229}{19}a^{7}-\frac{129}{19}a^{6}+\frac{1522}{95}a^{5}-\frac{809}{95}a^{4}+\frac{2056}{95}a^{3}-\frac{1067}{95}a^{2}-\frac{234}{95}a+\frac{223}{95}$
| 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: | \( 275.14971313898474 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 275.14971313898474 \cdot 1}{4\cdot\sqrt{47608675209216}}\cr\approx \mathstrut & 0.613402066642469 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 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 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 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 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 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 + 2*x^10 + 7*x^8 + 8*x^6 + 11*x^4 - 5*x^2 + 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
$C_6\times S_3$ (as 12T18):
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 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{11})\), 6.0.107811.1 |
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
| Galois closure: | deg 36 |
| Degree 18 siblings: | 18.6.174575864621452287114215424.1, 18.0.131161430970287217967104.1 |
| 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 | R | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
|
\(3\)
| 3.6.8.4 | $x^{6} + 18 x^{5} + 114 x^{4} + 344 x^{3} + 732 x^{2} + 744 x + 296$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ |
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(11\)
| 11.12.6.1 | $x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.36.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.396.6t1.b.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 11 $ | 6.6.558892224.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.396.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 11 $ | 6.6.558892224.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.99.6t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.36.6t1.b.b | $1$ | $ 2^{2} \cdot 3^{2}$ | 6.0.419904.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.99.6t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.891.3t2.b.a | $2$ | $ 3^{4} \cdot 11 $ | 3.1.891.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.14256.6t3.g.a | $2$ | $ 2^{4} \cdot 3^{4} \cdot 11 $ | 6.2.558892224.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.99.6t5.a.a | $2$ | $ 3^{2} \cdot 11 $ | 6.0.107811.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.99.6t5.a.b | $2$ | $ 3^{2} \cdot 11 $ | 6.0.107811.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1584.12t18.f.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 11 $ | 12.0.47608675209216.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.1584.12t18.f.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 11 $ | 12.0.47608675209216.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.