Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*x + 1)
gp: K = bnfinit(y^12 - 2*y^10 - 10*y^9 + 3*y^8 + 21*y^6 - 34*y^4 + 23*y^2 + 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^10 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^10 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*x + 1)
\( x^{12} - 2x^{10} - 10x^{9} + 3x^{8} + 21x^{6} - 34x^{4} + 23x^{2} + 10x + 1 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1776132919332864\)
\(\medspace = 2^{12}\cdot 3^{6}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.65\) | 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: | $2\cdot 3^{1/2}29^{1/2}\approx 18.65475810617763$ | ||
| Ramified primes: |
\(2\), \(3\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{45281}a^{11}-\frac{1950}{45281}a^{10}-\frac{1106}{45281}a^{9}-\frac{16798}{45281}a^{8}+\frac{17940}{45281}a^{7}+\frac{19213}{45281}a^{6}-\frac{17942}{45281}a^{5}-\frac{15313}{45281}a^{4}+\frac{20137}{45281}a^{3}-\frac{8523}{45281}a^{2}+\frac{1746}{45281}a-\frac{8615}{45281}$
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: | $7$ | 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: |
$\frac{428602}{45281}a^{11}-\frac{248888}{45281}a^{10}-\frac{711519}{45281}a^{9}-\frac{3871562}{45281}a^{8}+\frac{3530469}{45281}a^{7}-\frac{2064798}{45281}a^{6}+\frac{10192809}{45281}a^{5}-\frac{5904973}{45281}a^{4}-\frac{11120376}{45281}a^{3}+\frac{6499731}{45281}a^{2}+\frac{6047659}{45281}a+\frac{712130}{45281}$, $\frac{20166}{45281}a^{11}+\frac{25489}{45281}a^{10}-\frac{70625}{45281}a^{9}-\frac{227712}{45281}a^{8}-\frac{152993}{45281}a^{7}+\frac{342089}{45281}a^{6}+\frac{112661}{45281}a^{5}+\frac{784239}{45281}a^{4}-\frac{1491539}{45281}a^{3}-\frac{214547}{45281}a^{2}+\frac{1022681}{45281}a+\frac{239512}{45281}$, $a$, $\frac{732296}{45281}a^{11}-\frac{403113}{45281}a^{10}-\frac{1245997}{45281}a^{9}-\frac{6637493}{45281}a^{8}+\frac{5854959}{45281}a^{7}-\frac{3188380}{45281}a^{6}+\frac{17132189}{45281}a^{5}-\frac{9408570}{45281}a^{4}-\frac{19798986}{45281}a^{3}+\frac{10905829}{45281}a^{2}+\frac{10901940}{45281}a+\frac{1358434}{45281}$, $\frac{435987}{45281}a^{11}-\frac{250280}{45281}a^{10}-\frac{728749}{45281}a^{9}-\frac{3945414}{45281}a^{8}+\frac{3570444}{45281}a^{7}-\frac{2042166}{45281}a^{6}+\frac{10364469}{45281}a^{5}-\frac{5924821}{45281}a^{4}-\frac{11383121}{45281}a^{3}+\frac{6497966}{45281}a^{2}+\frac{6308470}{45281}a+\frac{800722}{45281}$, $\frac{1620030}{45281}a^{11}-\frac{935155}{45281}a^{10}-\frac{2700870}{45281}a^{9}-\frac{14642637}{45281}a^{8}+\frac{13312650}{45281}a^{7}-\frac{7677408}{45281}a^{6}+\frac{38463705}{45281}a^{5}-\frac{22194263}{45281}a^{4}-\frac{42263670}{45281}a^{3}+\frac{24380818}{45281}a^{2}+\frac{23188025}{45281}a+\frac{2849354}{45281}$, $\frac{154418}{45281}a^{11}-\frac{132293}{45281}a^{10}-\frac{212781}{45281}a^{9}-\frac{1349909}{45281}a^{8}+\frac{1642737}{45281}a^{7}-\frac{1245954}{45281}a^{6}+\frac{4161362}{45281}a^{5}-\frac{3425089}{45281}a^{4}-\frac{2828988}{45281}a^{3}+\frac{2704511}{45281}a^{2}+\frac{1459746}{45281}a+\frac{135272}{45281}$
| 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: | \( 957.10564041 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 957.10564041 \cdot 1}{2\cdot\sqrt{1776132919332864}}\cr\approx \mathstrut & 0.28315980476 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^10 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*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 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*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 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*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 - 10*x^9 + 3*x^8 + 21*x^6 - 34*x^4 + 23*x^2 + 10*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
$S_3\times D_6$ (as 12T37):
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 72 |
| The 18 conjugacy class representatives for $S_3\times D_6$ |
| Character table for $S_3\times D_6$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{87}) \), \(\Q(\sqrt{3}, \sqrt{29})\), 6.2.1453248.5 |
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 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.2111929749504.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |