Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729)
gp: K = bnfinit(y^12 - 8*y^10 + 23*y^8 - 56*y^6 + 207*y^4 - 648*y^2 + 729, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729)
\( x^{12} - 8x^{10} + 23x^{8} - 56x^{6} + 207x^{4} - 648x^{2} + 729 \)
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: |
\(284936905588473856\)
\(\medspace = 2^{24}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.48\) | 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: | $2^{9/4}19^{2/3}\approx 33.87036609855754$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{36}a^{8}-\frac{2}{9}a^{6}-\frac{1}{9}a^{4}+\frac{4}{9}a^{2}-\frac{1}{4}$, $\frac{1}{108}a^{9}-\frac{2}{27}a^{7}-\frac{1}{27}a^{5}+\frac{13}{27}a^{3}-\frac{1}{12}a$, $\frac{1}{2268}a^{10}+\frac{5}{1134}a^{8}-\frac{13}{81}a^{6}+\frac{7}{81}a^{4}-\frac{89}{252}a^{2}+\frac{5}{14}$, $\frac{1}{6804}a^{11}+\frac{5}{3402}a^{9}-\frac{13}{243}a^{7}+\frac{95}{486}a^{5}-\frac{89}{756}a^{3}-\frac{1}{21}a$
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: |
$\frac{23}{2268}a^{10}-\frac{37}{567}a^{8}+\frac{23}{162}a^{6}-\frac{28}{81}a^{4}+\frac{431}{252}a^{2}-\frac{30}{7}$, $\frac{17}{2268}a^{11}-\frac{103}{2268}a^{9}+\frac{11}{162}a^{7}-\frac{35}{162}a^{5}+\frac{683}{756}a^{3}-\frac{169}{84}a-1$, $\frac{8}{567}a^{10}-\frac{46}{567}a^{8}+\frac{23}{162}a^{6}-\frac{28}{81}a^{4}+\frac{23}{14}a^{2}-\frac{32}{7}$, $\frac{1}{567}a^{11}-\frac{1}{1134}a^{9}+\frac{1}{162}a^{7}-\frac{13}{162}a^{5}+\frac{47}{378}a^{3}+\frac{2}{21}a$, $\frac{1}{324}a^{11}+\frac{1}{567}a^{10}-\frac{5}{324}a^{9}+\frac{10}{567}a^{8}+\frac{13}{162}a^{7}-\frac{23}{162}a^{6}-\frac{61}{162}a^{5}+\frac{28}{81}a^{4}+\frac{31}{108}a^{3}-\frac{241}{126}a^{2}-\frac{1}{4}a+\frac{24}{7}$, $\frac{1}{972}a^{11}+\frac{11}{756}a^{10}-\frac{53}{972}a^{9}+\frac{5}{756}a^{8}+\frac{35}{243}a^{7}-\frac{5}{27}a^{6}+\frac{31}{243}a^{5}-\frac{16}{27}a^{4}+\frac{65}{36}a^{3}-\frac{221}{252}a^{2}-\frac{13}{4}a+\frac{85}{28}$, $\frac{1}{108}a^{11}-\frac{1}{378}a^{10}-\frac{1}{18}a^{9}+\frac{11}{378}a^{8}+\frac{4}{27}a^{7}+\frac{1}{54}a^{6}-\frac{7}{27}a^{5}+\frac{7}{27}a^{4}+\frac{167}{108}a^{3}-\frac{31}{63}a^{2}-\frac{23}{6}a+\frac{19}{14}$
| 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: | \( 18235.0066012 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 18235.0066012 \cdot 1}{2\cdot\sqrt{284936905588473856}}\cr\approx \mathstrut & 0.425932749236 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729)
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 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729, 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 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729);
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 - 8*x^10 + 23*x^8 - 56*x^6 + 207*x^4 - 648*x^2 + 729);
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
$C_2\times A_4$ (as 12T7):
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 24 |
| The 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.361.1, 6.2.66724352.1, 6.6.66724352.1, 6.2.8340544.2 |
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
| Galois closure: | deg 24 |
| Degree 6 sibling: | 6.2.66724352.1 |
| Degree 8 sibling: | 8.0.34162868224.3 |
| Degree 12 sibling: | 12.0.284936905588473856.21 |
| Minimal sibling: | 6.2.66724352.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{12}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.123 | $x^{12} + 6 x^{10} + 4 x^{9} + 50 x^{8} + 136 x^{7} + 224 x^{6} + 288 x^{5} + 140 x^{4} + 592 x^{3} + 664 x^{2} + 1776 x + 632$ | $4$ | $3$ | $24$ | $A_4 \times C_2$ | $[2, 2, 3]^{3}$ |
|
\(19\)
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |