Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8)
gp: K = bnfinit(y^12 - 12*y^10 + 72*y^8 - 212*y^6 + 324*y^4 - 72*y^2 + 8, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8)
\( x^{12} - 12x^{10} + 72x^{8} - 212x^{6} + 324x^{4} - 72x^{2} + 8 \)
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: |
\(5777633090469888\)
\(\medspace = 2^{27}\cdot 3^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.58\) | 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: | $2^{11/4}3^{4/3}\approx 29.106779845745038$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4964}a^{10}-\frac{14}{1241}a^{8}+\frac{27}{2482}a^{6}-\frac{53}{2482}a^{4}+\frac{6}{1241}a^{2}-\frac{282}{1241}$, $\frac{1}{4964}a^{11}-\frac{14}{1241}a^{9}+\frac{27}{2482}a^{7}-\frac{53}{2482}a^{5}+\frac{6}{1241}a^{3}-\frac{282}{1241}a$
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 $1$ (assuming GRH)
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: |
\( -\frac{42}{1241} a^{10} + \frac{981}{2482} a^{8} - \frac{5777}{2482} a^{6} + \frac{8175}{1241} a^{4} - \frac{12177}{1241} a^{2} + \frac{1459}{1241} \)
(order $4$)
| 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: |
$\frac{15}{1241}a^{10}-\frac{439}{2482}a^{8}+\frac{2861}{2482}a^{6}-\frac{9385}{2482}a^{4}+\frac{6565}{1241}a^{2}-\frac{787}{1241}$, $\frac{7}{1241}a^{10}-\frac{327}{4964}a^{8}+\frac{378}{1241}a^{6}-\frac{742}{1241}a^{4}+\frac{168}{1241}a^{2}+\frac{791}{1241}$, $\frac{27}{4964}a^{11}-\frac{271}{4964}a^{9}+\frac{729}{2482}a^{7}-\frac{1431}{2482}a^{5}+\frac{162}{1241}a^{3}+\frac{3555}{1241}a+1$, $\frac{339}{2482}a^{11}+\frac{111}{2482}a^{10}-\frac{2046}{1241}a^{9}-\frac{626}{1241}a^{8}+\frac{24511}{2482}a^{7}+\frac{7235}{2482}a^{6}-\frac{71923}{2482}a^{5}-\frac{20453}{2482}a^{4}+\frac{53708}{1241}a^{3}+\frac{16224}{1241}a^{2}-\frac{10010}{1241}a-\frac{4277}{1241}$, $\frac{141}{4964}a^{11}-\frac{43}{2482}a^{10}-\frac{1691}{4964}a^{9}+\frac{1093}{4964}a^{8}+\frac{2524}{1241}a^{7}-\frac{3563}{2482}a^{6}-\frac{14919}{2482}a^{5}+\frac{6002}{1241}a^{4}+\frac{12015}{1241}a^{3}-\frac{10444}{1241}a^{2}-\frac{5014}{1241}a+\frac{3155}{1241}$
| 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: | \( 848.12625378 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 848.12625378 \cdot 1}{4\cdot\sqrt{5777633090469888}}\cr\approx \mathstrut & 0.17163456222 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8)
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 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8, 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 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8);
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 - 12*x^10 + 72*x^8 - 212*x^6 + 324*x^4 - 72*x^2 + 8);
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
$C_3\times D_4$ (as 12T14):
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 15 conjugacy class representatives for $D_4 \times C_3$ |
| Character table for $D_4 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 4.0.512.1, 6.0.419904.1 |
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
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.6.46221064723759104.5 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.27.303 | $x^{12} + 6 x^{10} + 8 x^{9} + 570 x^{8} + 256 x^{7} + 4192 x^{6} + 8832 x^{5} + 9540 x^{4} + 3072 x^{3} + 600 x^{2} + 800 x + 1000$ | $4$ | $3$ | $27$ | $D_4 \times C_3$ | $[2, 3, 7/2]^{3}$ |
|
\(3\)
| 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |