Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*x + 1)
gp: K = bnfinit(y^12 - 2*y^11 - y^10 + 9*y^9 - 7*y^8 - 11*y^7 + 20*y^6 + y^5 - 19*y^4 + 8*y^3 + 6*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*x + 1)
\( x^{12} - 2x^{11} - x^{10} + 9x^{9} - 7x^{8} - 11x^{7} + 20x^{6} + x^{5} - 19x^{4} + 8x^{3} + 6x^{2} - 5x + 1 \)
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: |
\(213986328125\)
\(\medspace = 5^{9}\cdot 331^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(8.79\) | 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: | $5^{3/4}331^{2/3}\approx 159.99736578325135$ | ||
| Ramified primes: |
\(5\), \(331\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: |
\( 3 a^{11} - 5 a^{10} - 3 a^{9} + 22 a^{8} - 12 a^{7} - 27 a^{6} + 37 a^{5} + 13 a^{4} - 33 a^{3} + 4 a^{2} + 11 a - 3 \)
(order $10$)
| 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: |
$a^{10}-2a^{9}+6a^{7}-5a^{6}-5a^{5}+9a^{4}+2a^{3}-6a^{2}+1$, $4a^{11}-8a^{10}-3a^{9}+32a^{8}-24a^{7}-38a^{6}+62a^{5}+11a^{4}-58a^{3}+12a^{2}+20a-8$, $2a^{11}-4a^{10}-a^{9}+15a^{8}-12a^{7}-16a^{6}+29a^{5}+3a^{4}-25a^{3}+8a^{2}+8a-4$, $2a^{11}-2a^{10}-6a^{9}+17a^{8}+a^{7}-34a^{6}+24a^{5}+30a^{4}-35a^{3}-8a^{2}+17a-4$, $a^{11}-2a^{10}-a^{9}+9a^{8}-7a^{7}-11a^{6}+20a^{5}+a^{4}-19a^{3}+8a^{2}+6a-4$
| 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: | \( 16.35909998 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 16.35909998 \cdot 1}{10\cdot\sqrt{213986328125}}\cr\approx \mathstrut & 0.2175933425 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*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^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*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^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*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^11 - x^10 + 9*x^9 - 7*x^8 - 11*x^7 + 20*x^6 + x^5 - 19*x^4 + 8*x^3 + 6*x^2 - 5*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
$C_3\wr C_4$ (as 12T131):
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 324 |
| The 36 conjugacy class representatives for $C_3\wr C_4$ |
| Character table for $C_3\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
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 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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(331\)
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ |