Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*x + 1)
gp: K = bnfinit(y^12 - 3*y^11 - 12*y^10 + 47*y^9 - y^8 - 134*y^7 + 102*y^6 + 88*y^5 - 118*y^4 + 18*y^3 + 21*y^2 - 9*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*x + 1)
\( x^{12} - 3 x^{11} - 12 x^{10} + 47 x^{9} - x^{8} - 134 x^{7} + 102 x^{6} + 88 x^{5} - 118 x^{4} + \cdots + 1 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(54191813450093072\)
\(\medspace = 2^{4}\cdot 13^{4}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.80\) | 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: | $2^{2/3}13^{2/3}17^{3/4}\approx 73.47714215210142$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{11}-\frac{3}{13}a^{10}+\frac{3}{13}a^{9}-\frac{5}{13}a^{8}-\frac{5}{13}a^{7}-\frac{1}{2}a^{6}-\frac{1}{13}a^{5}+\frac{3}{26}a^{4}+\frac{3}{26}a^{3}+\frac{9}{26}a^{2}+\frac{7}{26}a-\frac{2}{13}$
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: | $11$ | 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: |
$a$, $a^{11}-2a^{10}-14a^{9}+33a^{8}+32a^{7}-102a^{6}+88a^{4}-30a^{3}-12a^{2}+9a-1$, $\frac{75}{26}a^{11}-\frac{82}{13}a^{10}-\frac{1045}{26}a^{9}+\frac{2695}{26}a^{8}+\frac{2279}{26}a^{7}-328a^{6}+\frac{331}{26}a^{5}+\frac{7843}{26}a^{4}-\frac{2375}{26}a^{3}-\frac{670}{13}a^{2}+\frac{629}{26}a-\frac{79}{26}$, $\frac{228}{13}a^{11}-\frac{1163}{26}a^{10}-\frac{5987}{26}a^{9}+\frac{18723}{26}a^{8}+\frac{7855}{26}a^{7}-\frac{4407}{2}a^{6}+\frac{10477}{13}a^{5}+\frac{24357}{13}a^{4}-\frac{31821}{26}a^{3}-\frac{2693}{13}a^{2}+\frac{6949}{26}a-\frac{561}{13}$, $\frac{217}{13}a^{11}-\frac{535}{13}a^{10}-\frac{2897}{13}a^{9}+\frac{8659}{13}a^{8}+\frac{4512}{13}a^{7}-2063a^{6}+\frac{7522}{13}a^{5}+\frac{23583}{13}a^{4}-\frac{12856}{13}a^{3}-\frac{3364}{13}a^{2}+\frac{2845}{13}a-\frac{361}{13}$, $\frac{103}{26}a^{11}-\frac{127}{13}a^{10}-\frac{1371}{26}a^{9}+\frac{4105}{26}a^{8}+\frac{2077}{26}a^{7}-486a^{6}+\frac{3811}{26}a^{5}+\frac{10917}{26}a^{4}-\frac{6425}{26}a^{3}-\frac{674}{13}a^{2}+\frac{1475}{26}a-\frac{243}{26}$, $2a^{11}-5a^{10}-26a^{9}+80a^{8}+31a^{7}-236a^{6}+102a^{5}+176a^{4}-148a^{3}+6a^{2}+31a-9$, $\frac{96}{13}a^{11}-\frac{251}{13}a^{10}-\frac{2501}{26}a^{9}+\frac{8051}{26}a^{8}+\frac{2981}{26}a^{7}-\frac{1883}{2}a^{6}+\frac{9691}{26}a^{5}+\frac{10272}{13}a^{4}-\frac{6966}{13}a^{3}-\frac{2211}{26}a^{2}+\frac{1491}{13}a-\frac{469}{26}$, $\frac{112}{13}a^{11}-\frac{282}{13}a^{10}-\frac{2959}{26}a^{9}+\frac{9109}{26}a^{8}+\frac{4143}{26}a^{7}-\frac{2163}{2}a^{6}+\frac{9523}{26}a^{5}+\frac{12257}{13}a^{4}-\frac{7568}{13}a^{3}-\frac{3223}{26}a^{2}+\frac{1681}{13}a-\frac{493}{26}$, $\frac{131}{13}a^{11}-\frac{675}{26}a^{10}-\frac{1710}{13}a^{9}+\frac{5424}{13}a^{8}+\frac{2096}{13}a^{7}-1271a^{6}+\frac{13087}{26}a^{5}+\frac{13861}{13}a^{4}-\frac{19325}{26}a^{3}-\frac{2699}{26}a^{2}+\frac{4239}{26}a-\frac{723}{26}$, $\frac{29}{2}a^{11}-\frac{71}{2}a^{10}-\frac{387}{2}a^{9}+\frac{1149}{2}a^{8}+\frac{603}{2}a^{7}-1773a^{6}+500a^{5}+\frac{3087}{2}a^{4}-851a^{3}-\frac{415}{2}a^{2}+185a-25$
| 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: | \( 29571.8190965 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 29571.8190965 \cdot 1}{2\cdot\sqrt{54191813450093072}}\cr\approx \mathstrut & 0.260160375607 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*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 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*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 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*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 - 3*x^11 - 12*x^10 + 47*x^9 - x^8 - 134*x^7 + 102*x^6 + 88*x^5 - 118*x^4 + 18*x^3 + 21*x^2 - 9*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
$C_3^2:C_{12}$ (as 12T73):
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 108 |
| The 18 conjugacy class representatives for $C_3^2:C_{12}$ |
| Character table for $C_3^2:C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1 |
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
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | 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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |