Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*x^2 - 6*x - 1)
gp: K = bnfinit(y^12 - 18*y^10 - 8*y^9 + 81*y^8 + 54*y^7 - 120*y^6 - 102*y^5 + 45*y^4 + 54*y^3 + 3*y^2 - 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*x^2 - 6*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*x^2 - 6*x - 1)
\( x^{12} - 18x^{10} - 8x^{9} + 81x^{8} + 54x^{7} - 120x^{6} - 102x^{5} + 45x^{4} + 54x^{3} + 3x^{2} - 6x - 1 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14003989580611584\)
\(\medspace = 2^{18}\cdot 3^{16}\cdot 17\cdot 73\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.16\) | 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: | $2^{3/2}3^{4/3}17^{1/2}73^{1/2}\approx 431.11442768403583$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1241}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1513}a^{11}+\frac{31}{1513}a^{10}-\frac{570}{1513}a^{9}+\frac{478}{1513}a^{8}-\frac{231}{1513}a^{7}+\frac{458}{1513}a^{6}+\frac{461}{1513}a^{5}+\frac{572}{1513}a^{4}-\frac{379}{1513}a^{3}+\frac{409}{1513}a^{2}+\frac{34}{89}a-\frac{244}{1513}$
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: |
$\frac{83100}{1513}a^{11}-\frac{47442}{1513}a^{10}-\frac{1468632}{1513}a^{9}+\frac{173493}{1513}a^{8}+\frac{6630810}{1513}a^{7}+\frac{703830}{1513}a^{6}-\frac{10370162}{1513}a^{5}-\frac{2563743}{1513}a^{4}+\frac{5201402}{1513}a^{3}+\frac{1529511}{1513}a^{2}-\frac{36662}{89}a-\frac{147448}{1513}$, $\frac{127494}{1513}a^{11}-\frac{72266}{1513}a^{10}-\frac{2253534}{1513}a^{9}+\frac{257215}{1513}a^{8}+\frac{10174356}{1513}a^{7}+\frac{1117637}{1513}a^{6}-\frac{15903920}{1513}a^{5}-\frac{3983761}{1513}a^{4}+\frac{7957322}{1513}a^{3}+\frac{2362807}{1513}a^{2}-\frac{55574}{89}a-\frac{226693}{1513}$, $a^{11}+18a^{9}+8a^{8}-81a^{7}-54a^{6}+120a^{5}+102a^{4}-45a^{3}-54a^{2}-3a+6$, $a^{11}+18a^{9}+8a^{8}-81a^{7}-54a^{6}+120a^{5}+102a^{4}-45a^{3}-54a^{2}-3a+5$, $\frac{102944}{1513}a^{11}+\frac{57147}{1513}a^{10}+\frac{1821053}{1513}a^{9}-\frac{187545}{1513}a^{8}-\frac{8230477}{1513}a^{7}-\frac{985209}{1513}a^{6}+\frac{12884282}{1513}a^{5}+\frac{3324540}{1513}a^{4}-\frac{6461978}{1513}a^{3}-\frac{1939998}{1513}a^{2}+\frac{45397}{89}a+\frac{182583}{1513}$, $\frac{42881}{1513}a^{11}+\frac{24824}{1513}a^{10}+\frac{757668}{1513}a^{9}-\frac{95826}{1513}a^{8}-\frac{3420993}{1513}a^{7}-\frac{332105}{1513}a^{6}+\frac{5352198}{1513}a^{5}+\frac{1265692}{1513}a^{4}-\frac{2687835}{1513}a^{3}-\frac{751594}{1513}a^{2}+\frac{19090}{89}a+\frac{71680}{1513}$, $\frac{42881}{1513}a^{11}+\frac{24824}{1513}a^{10}+\frac{757668}{1513}a^{9}-\frac{95826}{1513}a^{8}-\frac{3420993}{1513}a^{7}-\frac{332105}{1513}a^{6}+\frac{5352198}{1513}a^{5}+\frac{1265692}{1513}a^{4}-\frac{2687835}{1513}a^{3}-\frac{751594}{1513}a^{2}+\frac{19090}{89}a+\frac{70167}{1513}$, $\frac{144422}{1513}a^{11}-\frac{81587}{1513}a^{10}-\frac{2553667}{1513}a^{9}+\frac{287535}{1513}a^{8}+\frac{11538306}{1513}a^{7}+\frac{1276914}{1513}a^{6}-\frac{18061704}{1513}a^{5}-\frac{4515208}{1513}a^{4}+\frac{9062733}{1513}a^{3}+\frac{2665471}{1513}a^{2}-\frac{63684}{89}a-\frac{252356}{1513}$, $\frac{5973}{89}a^{11}+\frac{3339}{89}a^{10}+\frac{105647}{89}a^{9}-\frac{11277}{89}a^{8}-\frac{477489}{89}a^{7}-\frac{55577}{89}a^{6}+\frac{747618}{89}a^{5}+\frac{191148}{89}a^{4}-\frac{374994}{89}a^{3}-\frac{112581}{89}a^{2}+\frac{44505}{89}a+\frac{10628}{89}$, $\frac{18047}{1513}a^{11}+\frac{9431}{1513}a^{10}+\frac{319146}{1513}a^{9}-\frac{22035}{1513}a^{8}-\frac{1436808}{1513}a^{7}-\frac{223931}{1513}a^{6}+\frac{2224430}{1513}a^{5}+\frac{666035}{1513}a^{4}-\frac{1080742}{1513}a^{3}-\frac{382085}{1513}a^{2}+\frac{7177}{89}a+\frac{38463}{1513}$, $\frac{144520}{1513}a^{11}+\frac{83088}{1513}a^{10}+\frac{2553546}{1513}a^{9}-\frac{311684}{1513}a^{8}-\frac{11526259}{1513}a^{7}-\frac{1181089}{1513}a^{6}+\frac{18018039}{1513}a^{5}+\frac{4394093}{1513}a^{4}-\frac{9021052}{1513}a^{3}-\frac{2625364}{1513}a^{2}+\frac{62844}{89}a+\frac{252060}{1513}$
| 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: | \( 10781.91758130257 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 10781.91758130257 \cdot 1}{2\cdot\sqrt{14003989580611584}}\cr\approx \mathstrut & 0.186595029345050 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*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 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*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 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*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 - 18*x^10 - 8*x^9 + 81*x^8 + 54*x^7 - 120*x^6 - 102*x^5 + 45*x^4 + 54*x^3 + 3*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
$C_2\wr C_6$ (as 12T134):
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 384 |
| The 28 conjugacy class representatives for $C_2\wr C_6$ |
| Character table for $C_2\wr C_6$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 6.6.3359232.1 |
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 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(3\)
| 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |