Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109)
gp: K = bnfinit(y^12 - 2*y^11 + 6*y^10 - 14*y^9 + 39*y^8 - 72*y^7 + 160*y^6 - 218*y^5 + 338*y^4 - 316*y^3 + 338*y^2 - 176*y + 109, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109)
\( x^{12} - 2 x^{11} + 6 x^{10} - 14 x^{9} + 39 x^{8} - 72 x^{7} + 160 x^{6} - 218 x^{5} + 338 x^{4} + \cdots + 109 \)
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: |
\(45781510778228736\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 7^{6}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.46\) | 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: | $2\cdot 3^{1/2}7^{1/2}19^{2/3}\approx 65.25924479612236$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{14}a^{10}+\frac{1}{14}a^{9}-\frac{1}{7}a^{8}-\frac{3}{14}a^{7}+\frac{3}{14}a^{6}-\frac{1}{7}a^{5}-\frac{5}{14}a^{4}-\frac{1}{14}a^{3}+\frac{5}{14}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{870338}a^{11}+\frac{10470}{435169}a^{10}-\frac{8957}{124334}a^{9}-\frac{5388}{62167}a^{8}-\frac{157769}{870338}a^{7}+\frac{26623}{435169}a^{6}-\frac{36877}{435169}a^{5}+\frac{84565}{435169}a^{4}-\frac{115264}{435169}a^{3}+\frac{30014}{62167}a^{2}+\frac{68287}{870338}a+\frac{174756}{435169}$
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: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{512}{62167}a^{11}-\frac{5055}{124334}a^{10}+\frac{2105}{17762}a^{9}-\frac{2211}{8881}a^{8}+\frac{39372}{62167}a^{7}-\frac{91528}{62167}a^{6}+\frac{381811}{124334}a^{5}-\frac{314906}{62167}a^{4}+\frac{857765}{124334}a^{3}-\frac{121063}{17762}a^{2}+\frac{609683}{124334}a-\frac{243799}{124334}$, $\frac{10235}{435169}a^{11}+\frac{31251}{435169}a^{10}-\frac{5213}{62167}a^{9}+\frac{28201}{124334}a^{8}-\frac{164391}{435169}a^{7}+\frac{887226}{435169}a^{6}-\frac{1221649}{435169}a^{5}+\frac{7408743}{870338}a^{4}-\frac{3262613}{435169}a^{3}+\frac{1640985}{124334}a^{2}-\frac{2885818}{435169}a+\frac{5616475}{870338}$, $\frac{11359}{435169}a^{11}-\frac{55649}{435169}a^{10}+\frac{22989}{124334}a^{9}-\frac{4755}{8881}a^{8}+\frac{1285583}{870338}a^{7}-\frac{1431270}{435169}a^{6}+\frac{2290985}{435169}a^{5}-\frac{4226487}{435169}a^{4}+\frac{8578523}{870338}a^{3}-\frac{771132}{62167}a^{2}+\frac{2874056}{435169}a-\frac{2861828}{435169}$, $\frac{19059}{435169}a^{11}-\frac{95527}{870338}a^{10}+\frac{16881}{62167}a^{9}-\frac{13321}{17762}a^{8}+\frac{1626679}{870338}a^{7}-\frac{3167705}{870338}a^{6}+\frac{6737305}{870338}a^{5}-\frac{5256642}{435169}a^{4}+\frac{6671442}{435169}a^{3}-\frac{1078012}{62167}a^{2}+\frac{11656805}{870338}a-\frac{7584893}{870338}$, $\frac{49751}{870338}a^{11}+\frac{53521}{870338}a^{10}+\frac{1115}{62167}a^{9}-\frac{3165}{62167}a^{8}+\frac{113201}{435169}a^{7}+\frac{612441}{435169}a^{6}-\frac{57290}{435169}a^{5}+\frac{3087773}{435169}a^{4}-\frac{4098221}{870338}a^{3}+\frac{1486761}{124334}a^{2}-\frac{4556037}{870338}a+\frac{7983819}{870338}$
| 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: | \( 678.2219033596916 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 678.2219033596916 \cdot 2}{2\cdot\sqrt{45781510778228736}}\cr\approx \mathstrut & 0.195032039083292 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109)
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^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109, 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^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109);
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^11 + 6*x^10 - 14*x^9 + 39*x^8 - 72*x^7 + 160*x^6 - 218*x^5 + 338*x^4 - 316*x^3 + 338*x^2 - 176*x + 109);
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
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}, \sqrt{-7})\), 6.0.213966144.1 |
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: | 18.6.1343577325387856327724233392128.1, deg 18 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(19\)
| 19.6.4.2 | $x^{6} - 342 x^{3} + 722$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.84.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | \(\Q(\sqrt{-21}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.1596.6t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 6.0.77241777984.6 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.133.6t1.j.a | $1$ | $ 7 \cdot 19 $ | 6.0.44700103.1 | $C_6$ (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) | $0$ | $1$ | |
| 1.228.6t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 19 $ | 6.6.225194688.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.1596.6t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 6.0.77241777984.6 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.228.6t1.a.b | $1$ | $ 2^{2} \cdot 3 \cdot 19 $ | 6.6.225194688.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.133.6t1.j.b | $1$ | $ 7 \cdot 19 $ | 6.0.44700103.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.30324.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19^{2}$ | 3.1.30324.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.30324.6t3.c.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19^{2}$ | 6.2.11034539712.4 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.1596.6t5.c.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 6.0.213966144.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1596.6t5.c.b | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 6.0.213966144.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1596.12t18.b.a | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 12.0.45781510778228736.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.1596.12t18.b.b | $2$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 19 $ | 12.0.45781510778228736.1 | $C_6\times S_3$ (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 *.