Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 1)
gp: K = bnfinit(y^12 - 2*y^11 - 2*y^10 + 17*y^8 - 10*y^7 - 8*y^6 - 20*y^5 + 31*y^4 - 2*y^3 - 3*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 1)
\( x^{12} - 2x^{11} - 2x^{10} + 17x^{8} - 10x^{7} - 8x^{6} - 20x^{5} + 31x^{4} - 2x^{3} - 3x^{2} - 2x + 1 \)
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: |
\(5596214759424\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 37^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.54\) | 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: | $2\cdot 3^{1/2}37^{2/3}\approx 38.464353655440554$ | ||
| Ramified primes: |
\(2\), \(3\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{18469}a^{11}+\frac{2422}{18469}a^{10}-\frac{2216}{18469}a^{9}+\frac{2895}{18469}a^{8}-\frac{723}{18469}a^{7}+\frac{1993}{18469}a^{6}-\frac{714}{1679}a^{5}+\frac{3423}{18469}a^{4}+\frac{4802}{18469}a^{3}+\frac{416}{1679}a^{2}-\frac{7648}{18469}a+\frac{4122}{18469}$
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: |
\( \frac{7290}{18469} a^{11} - \frac{18453}{18469} a^{10} - \frac{12734}{18469} a^{9} + \frac{12952}{18469} a^{8} + \frac{140747}{18469} a^{7} - \frac{116947}{18469} a^{6} - \frac{8555}{1679} a^{5} - \frac{145701}{18469} a^{4} + \frac{303329}{18469} a^{3} + \frac{366}{1679} a^{2} - \frac{32947}{18469} a - \frac{18152}{18469} \)
(order $12$)
| 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: |
$\frac{14253}{18469}a^{11}-\frac{16264}{18469}a^{10}-\frac{39596}{18469}a^{9}-\frac{34249}{18469}a^{8}+\frac{203942}{18469}a^{7}+\frac{19376}{18469}a^{6}-\frac{6939}{1679}a^{5}-\frac{321052}{18469}a^{4}+\frac{181482}{18469}a^{3}+\frac{9094}{1679}a^{2}-\frac{2906}{18469}a-\frac{17492}{18469}$, $\frac{2705}{18469}a^{11}-\frac{4985}{18469}a^{10}-\frac{10324}{18469}a^{9}+\frac{119}{18469}a^{8}+\frac{57406}{18469}a^{7}-\frac{1883}{18469}a^{6}-\frac{5557}{1679}a^{5}-\frac{86099}{18469}a^{4}+\frac{61110}{18469}a^{3}+\frac{5387}{1679}a^{2}-\frac{2560}{18469}a-\frac{5266}{18469}$, $\frac{17492}{18469}a^{11}-\frac{20731}{18469}a^{10}-\frac{51248}{18469}a^{9}-\frac{39596}{18469}a^{8}+\frac{263115}{18469}a^{7}+\frac{29022}{18469}a^{6}-\frac{10960}{1679}a^{5}-\frac{426169}{18469}a^{4}+\frac{221200}{18469}a^{3}+\frac{13318}{1679}a^{2}+\frac{47558}{18469}a-\frac{19421}{18469}$, $\frac{22760}{18469}a^{11}-\frac{42183}{18469}a^{10}-\frac{52728}{18469}a^{9}-\frac{7192}{18469}a^{8}+\frac{388248}{18469}a^{7}-\frac{165405}{18469}a^{6}-\frac{19747}{1679}a^{5}-\frac{493425}{18469}a^{4}+\frac{621924}{18469}a^{3}+\frac{6995}{1679}a^{2}-\frac{53562}{18469}a-\frac{61207}{18469}$, $\frac{7290}{18469}a^{11}-\frac{18453}{18469}a^{10}-\frac{12734}{18469}a^{9}+\frac{12952}{18469}a^{8}+\frac{140747}{18469}a^{7}-\frac{116947}{18469}a^{6}-\frac{8555}{1679}a^{5}-\frac{145701}{18469}a^{4}+\frac{303329}{18469}a^{3}+\frac{366}{1679}a^{2}-\frac{32947}{18469}a-\frac{36621}{18469}$
| 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: | \( 91.03461538825627 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 91.03461538825627 \cdot 1}{12\cdot\sqrt{5596214759424}}\cr\approx \mathstrut & 0.197313744996243 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 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^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 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^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 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^11 - 2*x^10 + 17*x^8 - 10*x^7 - 8*x^6 - 20*x^5 + 31*x^4 - 2*x^3 - 3*x^2 - 2*x + 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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.87616.1 |
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: | 18.6.33966586417892403297890598912.2, 18.0.530727912779568801529540608.1 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.3.2.3 | $x^{3} + 111$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.2.3 | $x^{3} + 111$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
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) | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.148.6t1.b.a | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.111.6t1.b.a | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.444.6t1.b.a | $1$ | $ 2^{2} \cdot 3 \cdot 37 $ | 6.6.3238550208.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.148.6t1.b.b | $1$ | $ 2^{2} \cdot 37 $ | 6.0.119946304.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.444.6t1.b.b | $1$ | $ 2^{2} \cdot 3 \cdot 37 $ | 6.6.3238550208.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.111.6t1.b.b | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.5476.3t2.a.a | $2$ | $ 2^{2} \cdot 37^{2}$ | 3.1.5476.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.49284.6t3.c.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 37^{2}$ | 6.2.3238550208.12 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.1332.12t18.b.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 37 $ | 12.0.5596214759424.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.148.6t5.b.a | $2$ | $ 2^{2} \cdot 37 $ | 6.0.87616.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1332.12t18.b.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 37 $ | 12.0.5596214759424.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.148.6t5.b.b | $2$ | $ 2^{2} \cdot 37 $ | 6.0.87616.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.