Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64)
gp: K = bnfinit(y^12 - y^11 + 8*y^10 - y^9 + 23*y^8 + 2*y^7 + 41*y^6 - 14*y^5 + 8*y^4 - 40*y^3 - 48*y^2 + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64)
\( x^{12} - x^{11} + 8x^{10} - x^{9} + 23x^{8} + 2x^{7} + 41x^{6} - 14x^{5} + 8x^{4} - 40x^{3} - 48x^{2} + 64 \)
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: |
\(11177126654841\)
\(\medspace = 3^{6}\cdot 7^{6}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.23\) | 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: | $3^{1/2}7^{1/2}19^{2/3}\approx 32.62962239806118$ | ||
| Ramified primes: |
\(3\), \(7\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{3}{8}a^{5}-\frac{7}{16}a^{4}-\frac{3}{8}a^{3}$, $\frac{1}{66746272}a^{11}+\frac{1791543}{66746272}a^{10}+\frac{8171}{4171642}a^{9}+\frac{6627935}{66746272}a^{8}+\frac{8646591}{66746272}a^{7}+\frac{3217829}{33373136}a^{6}+\frac{13450713}{66746272}a^{5}+\frac{2049077}{33373136}a^{4}+\frac{4507}{8343284}a^{3}-\frac{1810109}{8343284}a^{2}+\frac{1368335}{4171642}a+\frac{155538}{2085821}$
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: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{269}{34264} a^{11} + \frac{469}{68528} a^{10} - \frac{4825}{68528} a^{9} - \frac{31}{8566} a^{8} - \frac{18281}{68528} a^{7} - \frac{2521}{68528} a^{6} - \frac{23359}{34264} a^{5} + \frac{171}{68528} a^{4} - \frac{4871}{8566} a^{3} + \frac{4449}{17132} a^{2} - \frac{97}{8566} a + \frac{3620}{4283} \)
(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: |
$\frac{26821}{8343284}a^{11}-\frac{41787}{4171642}a^{10}+\frac{205155}{8343284}a^{9}-\frac{105433}{2085821}a^{8}+\frac{295147}{8343284}a^{7}-\frac{1333637}{8343284}a^{6}+\frac{29517}{4171642}a^{5}-\frac{501656}{2085821}a^{4}-\frac{190742}{2085821}a^{3}+\frac{3003035}{8343284}a^{2}+\frac{370160}{2085821}a+\frac{2427405}{2085821}$, $\frac{142149}{16686568}a^{11}-\frac{183267}{16686568}a^{10}+\frac{1412375}{16686568}a^{9}-\frac{73035}{2085821}a^{8}+\frac{5038315}{16686568}a^{7}+\frac{30831}{16686568}a^{6}+\frac{5233457}{8343284}a^{5}-\frac{56768}{2085821}a^{4}+\frac{12977073}{16686568}a^{3}-\frac{391502}{2085821}a^{2}-\frac{197729}{4171642}a+\frac{1560069}{2085821}$, $\frac{357201}{16686568}a^{11}-\frac{807345}{33373136}a^{10}+\frac{5424597}{33373136}a^{9}+\frac{35337}{4171642}a^{8}+\frac{14299397}{33373136}a^{7}+\frac{7294581}{33373136}a^{6}+\frac{15408579}{16686568}a^{5}+\frac{6397857}{33373136}a^{4}+\frac{1388011}{4171642}a^{3}-\frac{174717}{8343284}a^{2}-\frac{5475843}{4171642}a-\frac{2567714}{2085821}$, $\frac{1619353}{66746272}a^{11}-\frac{2867369}{66746272}a^{10}+\frac{3397219}{16686568}a^{9}-\frac{9320509}{66746272}a^{8}+\frac{33687711}{66746272}a^{7}-\frac{9492489}{33373136}a^{6}+\frac{66417373}{66746272}a^{5}-\frac{28492459}{33373136}a^{4}+\frac{6530047}{16686568}a^{3}-\frac{1636999}{4171642}a^{2}-\frac{320749}{4171642}a-\frac{302120}{2085821}$, $\frac{1302657}{66746272}a^{11}-\frac{684161}{66746272}a^{10}+\frac{2348957}{16686568}a^{9}+\frac{3625155}{66746272}a^{8}+\frac{25459447}{66746272}a^{7}+\frac{7358707}{33373136}a^{6}+\frac{43178733}{66746272}a^{5}-\frac{5665963}{33373136}a^{4}-\frac{7279495}{16686568}a^{3}-\frac{2966533}{2085821}a^{2}-\frac{5485848}{2085821}a-\frac{3987474}{2085821}$
| 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: | \( 58.233702693092496 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 58.233702693092496 \cdot 1}{6\cdot\sqrt{11177126654841}}\cr\approx \mathstrut & 0.178622958625082 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64)
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 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64, 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 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64);
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 - x^11 + 8*x^10 - x^9 + 23*x^8 + 2*x^7 + 41*x^6 - 14*x^5 + 8*x^4 - 40*x^3 - 48*x^2 + 64);
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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.9747.1 |
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: | 18.6.1757991877014292604100845541.1, 18.0.65110810259788614966697983.1 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.133.6t1.j.a | $1$ | $ 7 \cdot 19 $ | 6.0.44700103.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.399.6t1.f.a | $1$ | $ 3 \cdot 7 \cdot 19 $ | 6.6.1206902781.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.399.6t1.f.b | $1$ | $ 3 \cdot 7 \cdot 19 $ | 6.6.1206902781.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.133.6t1.j.b | $1$ | $ 7 \cdot 19 $ | 6.0.44700103.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.1083.3t2.b.a | $2$ | $ 3 \cdot 19^{2}$ | 3.1.1083.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.53067.6t3.b.a | $2$ | $ 3 \cdot 7^{2} \cdot 19^{2}$ | 6.2.1206902781.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.57.6t5.a.a | $2$ | $ 3 \cdot 19 $ | 6.0.9747.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2793.12t18.d.a | $2$ | $ 3 \cdot 7^{2} \cdot 19 $ | 12.0.11177126654841.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.57.6t5.a.b | $2$ | $ 3 \cdot 19 $ | 6.0.9747.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2793.12t18.d.b | $2$ | $ 3 \cdot 7^{2} \cdot 19 $ | 12.0.11177126654841.1 | $C_6\times S_3$ (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 *.