Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*x^2 - 5*x + 1)
gp: K = bnfinit(y^12 - 4*y^11 + 9*y^10 - 13*y^9 + 17*y^8 - 21*y^7 + 25*y^6 - 27*y^5 + 26*y^4 - 20*y^3 + 12*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*x^2 - 5*x + 1)
\( x^{12} - 4 x^{11} + 9 x^{10} - 13 x^{9} + 17 x^{8} - 21 x^{7} + 25 x^{6} - 27 x^{5} + 26 x^{4} + \cdots + 1 \)
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: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(63853279857\)
\(\medspace = 3^{6}\cdot 23^{4}\cdot 313\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(7.95\) | 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: | $3^{1/2}23^{1/2}313^{1/2}\approx 146.9591780053223$ | ||
| Ramified primes: |
\(3\), \(23\), \(313\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{313}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17}a^{11}+\frac{3}{17}a^{10}-\frac{4}{17}a^{9}-\frac{7}{17}a^{8}+\frac{2}{17}a^{7}-\frac{7}{17}a^{6}-\frac{7}{17}a^{5}-\frac{8}{17}a^{4}+\frac{4}{17}a^{3}+\frac{8}{17}a^{2}-\frac{5}{17}$
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: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{65}{17} a^{11} - \frac{213}{17} a^{10} + \frac{420}{17} a^{9} - \frac{506}{17} a^{8} + \frac{674}{17} a^{7} - \frac{812}{17} a^{6} + \frac{956}{17} a^{5} - \frac{962}{17} a^{4} + \frac{872}{17} a^{3} - \frac{568}{17} a^{2} + 17 a - \frac{70}{17} \)
(order $6$)
| 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: |
$\frac{32}{17}a^{11}-\frac{108}{17}a^{10}+\frac{229}{17}a^{9}-\frac{292}{17}a^{8}+\frac{387}{17}a^{7}-\frac{445}{17}a^{6}+\frac{558}{17}a^{5}-\frac{562}{17}a^{4}+\frac{519}{17}a^{3}-\frac{339}{17}a^{2}+11a-\frac{41}{17}$, $\frac{47}{17}a^{11}-\frac{165}{17}a^{10}+\frac{339}{17}a^{9}-\frac{431}{17}a^{8}+\frac{553}{17}a^{7}-\frac{669}{17}a^{6}+\frac{793}{17}a^{5}-\frac{818}{17}a^{4}+\frac{732}{17}a^{3}-\frac{491}{17}a^{2}+15a-\frac{65}{17}$, $\frac{18}{17}a^{11}-\frac{48}{17}a^{10}+\frac{81}{17}a^{9}-\frac{75}{17}a^{8}+\frac{121}{17}a^{7}-\frac{143}{17}a^{6}+\frac{163}{17}a^{5}-\frac{144}{17}a^{4}+\frac{140}{17}a^{3}-\frac{77}{17}a^{2}+2a-\frac{22}{17}$, $\frac{80}{17}a^{11}-\frac{287}{17}a^{10}+\frac{598}{17}a^{9}-\frac{781}{17}a^{8}+\frac{1010}{17}a^{7}-\frac{1223}{17}a^{6}+\frac{1446}{17}a^{5}-\frac{1507}{17}a^{4}+\frac{1391}{17}a^{3}-\frac{941}{17}a^{2}+30a-\frac{145}{17}$, $\frac{55}{17}a^{11}-\frac{192}{17}a^{10}+\frac{392}{17}a^{9}-\frac{487}{17}a^{8}+\frac{620}{17}a^{7}-\frac{742}{17}a^{6}+\frac{890}{17}a^{5}-\frac{899}{17}a^{4}+\frac{815}{17}a^{3}-\frac{512}{17}a^{2}+14a-\frac{54}{17}$
| 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: | \( 4.5780204292 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4.5780204292 \cdot 1}{6\cdot\sqrt{63853279857}}\cr\approx \mathstrut & 0.18578646918 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*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 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*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 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*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 - 4*x^11 + 9*x^10 - 13*x^9 + 17*x^8 - 21*x^7 + 25*x^6 - 27*x^5 + 26*x^4 - 20*x^3 + 12*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
$C_2\wr D_6$ (as 12T193):
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 768 |
| The 38 conjugacy class representatives for $C_2\wr D_6$ |
| Character table for $C_2\wr D_6$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1 |
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: | 12.2.740225059083.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(23\)
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(313\)
| $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |