Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*x + 1)
gp: K = bnfinit(y^12 - 4*y^11 + 9*y^10 - 14*y^9 + 16*y^8 - 14*y^7 + 10*y^6 - 6*y^5 + 6*y^4 - 6*y^3 + 6*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*x + 1)
\( x^{12} - 4x^{11} + 9x^{10} - 14x^{9} + 16x^{8} - 14x^{7} + 10x^{6} - 6x^{5} + 6x^{4} - 6x^{3} + 6x^{2} - 4x + 1 \)
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: |
\(63568613376\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 61\cdot 349\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(7.95\) | 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: | $2\cdot 3^{1/2}61^{1/2}349^{1/2}\approx 505.4384235492984$ | ||
| Ramified primes: |
\(2\), \(3\), \(61\), \(349\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{21289}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}+\frac{3}{11}a^{10}-\frac{3}{11}a^{9}-\frac{2}{11}a^{8}+\frac{2}{11}a^{7}-\frac{1}{11}a^{5}-\frac{2}{11}a^{4}+\frac{3}{11}a^{3}+\frac{4}{11}a^{2}+\frac{1}{11}a+\frac{3}{11}$
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: |
\( -\frac{26}{11} a^{11} + \frac{87}{11} a^{10} - \frac{175}{11} a^{9} + \frac{250}{11} a^{8} - \frac{261}{11} a^{7} + 19 a^{6} - \frac{139}{11} a^{5} + \frac{74}{11} a^{4} - \frac{111}{11} a^{3} + \frac{94}{11} a^{2} - \frac{103}{11} a + \frac{54}{11} \)
(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: |
$\frac{82}{11}a^{11}-\frac{282}{11}a^{10}+\frac{579}{11}a^{9}-\frac{824}{11}a^{8}+\frac{857}{11}a^{7}-62a^{6}+\frac{457}{11}a^{5}-\frac{252}{11}a^{4}+\frac{356}{11}a^{3}-\frac{288}{11}a^{2}+\frac{324}{11}a-\frac{150}{11}$, $\frac{30}{11}a^{11}-\frac{108}{11}a^{10}+\frac{229}{11}a^{9}-\frac{335}{11}a^{8}+\frac{357}{11}a^{7}-26a^{6}+\frac{190}{11}a^{5}-\frac{104}{11}a^{4}+\frac{134}{11}a^{3}-\frac{122}{11}a^{2}+\frac{129}{11}a-\frac{64}{11}$, $5a^{11}-17a^{10}+35a^{9}-50a^{8}+52a^{7}-41a^{6}+27a^{5}-15a^{4}+22a^{3}-18a^{2}+20a-9$, $\frac{70}{11}a^{11}-\frac{241}{11}a^{10}+\frac{494}{11}a^{9}-\frac{701}{11}a^{8}+\frac{723}{11}a^{7}-52a^{6}+\frac{381}{11}a^{5}-\frac{206}{11}a^{4}+\frac{298}{11}a^{3}-\frac{248}{11}a^{2}+\frac{279}{11}a-\frac{131}{11}$, $\frac{53}{11}a^{11}-\frac{182}{11}a^{10}+\frac{380}{11}a^{9}-\frac{546}{11}a^{8}+\frac{568}{11}a^{7}-41a^{6}+\frac{299}{11}a^{5}-\frac{161}{11}a^{4}+\frac{236}{11}a^{3}-\frac{195}{11}a^{2}+\frac{229}{11}a-\frac{105}{11}$
| 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: | \( 9.18542221354 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 9.18542221354 \cdot 1}{12\cdot\sqrt{63568613376}}\cr\approx \mathstrut & 0.186799503979 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*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 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*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 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*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 - 4*x^11 + 9*x^10 - 14*x^9 + 16*x^8 - 14*x^7 + 10*x^6 - 6*x^5 + 6*x^4 - 6*x^3 + 6*x^2 - 4*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
$S_3\wr C_2^2$ (as 12T261):
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 5184 |
| The 45 conjugacy class representatives for $S_3\wr C_2^2$ |
| Character table for $S_3\wr C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\) |
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
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 61.3.0.1 | $x^{3} + 7 x + 59$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(349\)
| $\Q_{349}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |