Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1)
gp: K = bnfinit(y^12 - 2*y^10 - 4*y^9 - 9*y^8 + 2*y^7 + 20*y^6 + 22*y^5 + 13*y^4 - 2*y^3 - 9*y^2 - 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*x - 1)
\( x^{12} - 2x^{10} - 4x^{9} - 9x^{8} + 2x^{7} + 20x^{6} + 22x^{5} + 13x^{4} - 2x^{3} - 9x^{2} - 6x - 1 \)
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: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-122023936000000\)
\(\medspace = -\,2^{18}\cdot 5^{6}\cdot 31^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.92\) | 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: | $2^{3/2}5^{1/2}31^{5/6}\approx 110.61942767882832$ | ||
| Ramified primes: |
\(2\), \(5\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{325}a^{11}+\frac{21}{325}a^{10}-\frac{146}{325}a^{9}-\frac{3}{65}a^{8}+\frac{66}{325}a^{7}+\frac{153}{325}a^{6}+\frac{113}{325}a^{5}+\frac{24}{65}a^{4}+\frac{128}{325}a^{3}-\frac{44}{325}a^{2}+\frac{107}{325}a+\frac{96}{325}$
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: | $8$ | 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: |
$\frac{22}{65}a^{11}-\frac{19}{65}a^{10}-\frac{14}{65}a^{9}-\frac{96}{65}a^{8}-\frac{134}{65}a^{7}+\frac{129}{65}a^{6}+\frac{276}{65}a^{5}+\frac{417}{65}a^{4}+\frac{138}{65}a^{3}-\frac{149}{65}a^{2}-\frac{168}{65}a-\frac{72}{65}$, $a$, $\frac{226}{325}a^{11}-\frac{129}{325}a^{10}-\frac{496}{325}a^{9}-\frac{93}{65}a^{8}-\frac{1659}{325}a^{7}+\frac{1753}{325}a^{6}+\frac{4088}{325}a^{5}+\frac{289}{65}a^{4}+\frac{653}{325}a^{3}-\frac{1819}{325}a^{2}-\frac{1168}{325}a-\frac{79}{325}$, $\frac{236}{325}a^{11}-\frac{179}{325}a^{10}-\frac{201}{325}a^{9}-\frac{35}{13}a^{8}-\frac{1584}{325}a^{7}+\frac{1138}{325}a^{6}+\frac{2943}{325}a^{5}+\frac{698}{65}a^{4}+\frac{2128}{325}a^{3}+\frac{81}{325}a^{2}-\frac{1268}{325}a-\frac{484}{325}$, $\frac{212}{325}a^{11}-\frac{293}{325}a^{10}-\frac{142}{325}a^{9}-\frac{18}{13}a^{8}-\frac{1153}{325}a^{7}+\frac{2146}{325}a^{6}+\frac{1856}{325}a^{5}+\frac{96}{65}a^{4}+\frac{876}{325}a^{3}-\frac{1073}{325}a^{2}-\frac{456}{325}a+\frac{72}{325}$, $\frac{141}{325}a^{11}-\frac{29}{325}a^{10}-\frac{241}{325}a^{9}-\frac{111}{65}a^{8}-\frac{1159}{325}a^{7}+\frac{318}{325}a^{6}+\frac{2608}{325}a^{5}+\frac{110}{13}a^{4}+\frac{1603}{325}a^{3}+\frac{231}{325}a^{2}-\frac{968}{325}a-\frac{374}{325}$, $\frac{357}{325}a^{11}-\frac{498}{325}a^{10}-\frac{187}{325}a^{9}-\frac{40}{13}a^{8}-\frac{1658}{325}a^{7}+\frac{3531}{325}a^{6}+\frac{3291}{325}a^{5}+\frac{391}{65}a^{4}-\frac{64}{325}a^{3}-\frac{2253}{325}a^{2}-\frac{1191}{325}a+\frac{17}{325}$, $\frac{9}{25}a^{11}-\frac{11}{25}a^{10}-\frac{14}{25}a^{9}-\frac{2}{5}a^{8}-\frac{56}{25}a^{7}+\frac{102}{25}a^{6}+\frac{117}{25}a^{5}-\frac{4}{5}a^{4}+\frac{27}{25}a^{3}-\frac{71}{25}a^{2}-\frac{37}{25}a+\frac{14}{25}$
| 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: | \( 280.158137734 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 280.158137734 \cdot 1}{2\cdot\sqrt{122023936000000}}\cr\approx \mathstrut & 0.201312299938 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*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^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*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^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*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^10 - 4*x^9 - 9*x^8 + 2*x^7 + 20*x^6 + 22*x^5 + 13*x^4 - 2*x^3 - 9*x^2 - 6*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 $S_3\wr C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\) |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | R | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |