Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*x - 1)
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 - 7*y^8 + 6*y^7 - 11*y^6 + 8*y^5 + 21*y^4 - 18*y^3 - 6*y^2 + 7*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*x - 1)
\( x^{12} - 3x^{11} + 2x^{10} - 7x^{8} + 6x^{7} - 11x^{6} + 8x^{5} + 21x^{4} - 18x^{3} - 6x^{2} + 7x - 1 \)
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: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-40634924000000\)
\(\medspace = -\,2^{8}\cdot 5^{6}\cdot 11\cdot 31^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.62\) | 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: | $2^{2/3}5^{1/2}11^{1/2}31^{2/3}\approx 116.17405213303951$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{1627}a^{11}+\frac{116}{1627}a^{10}+\frac{790}{1627}a^{9}-\frac{356}{1627}a^{8}-\frac{69}{1627}a^{7}-\frac{70}{1627}a^{6}-\frac{206}{1627}a^{5}-\frac{101}{1627}a^{4}-\frac{609}{1627}a^{3}+\frac{726}{1627}a^{2}+\frac{157}{1627}a+\frac{793}{1627}$
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: | $8$ | 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: |
$\frac{5331}{1627}a^{11}-\frac{12880}{1627}a^{10}+\frac{4068}{1627}a^{9}-\frac{754}{1627}a^{8}-\frac{34304}{1627}a^{7}+\frac{9175}{1627}a^{6}-\frac{56906}{1627}a^{5}+\frac{14749}{1627}a^{4}+\frac{106668}{1627}a^{3}-\frac{18224}{1627}a^{2}-\frac{35105}{1627}a+\frac{3791}{1627}$, $\frac{3113}{1627}a^{11}-\frac{6594}{1627}a^{10}-\frac{754}{1627}a^{9}+\frac{3013}{1627}a^{8}-\frac{22811}{1627}a^{7}+\frac{1735}{1627}a^{6}-\frac{27899}{1627}a^{5}-\frac{5283}{1627}a^{4}+\frac{77734}{1627}a^{3}-\frac{3119}{1627}a^{2}-\frac{33526}{1627}a+\frac{5331}{1627}$, $\frac{17658}{1627}a^{11}-\frac{40740}{1627}a^{10}+\frac{9684}{1627}a^{9}-\frac{1147}{1627}a^{8}-\frac{116923}{1627}a^{7}+\frac{18357}{1627}a^{6}-\frac{193189}{1627}a^{5}+\frac{17631}{1627}a^{4}+\frac{345672}{1627}a^{3}-\frac{48235}{1627}a^{2}-\frac{115619}{1627}a+\frac{21983}{1627}$, $\frac{3293}{1627}a^{11}-\frac{6865}{1627}a^{10}-\frac{103}{1627}a^{9}+\frac{759}{1627}a^{8}-\frac{22215}{1627}a^{7}-\frac{1103}{1627}a^{6}-\frac{34066}{1627}a^{5}-\frac{5566}{1627}a^{4}+\frac{68988}{1627}a^{3}+\frac{3909}{1627}a^{2}-\frac{26417}{1627}a+\frac{1641}{1627}$, $\frac{9749}{1627}a^{11}-\frac{22659}{1627}a^{10}+\frac{4373}{1627}a^{9}+\frac{3001}{1627}a^{8}-\frac{67437}{1627}a^{7}+\frac{13926}{1627}a^{6}-\frac{99823}{1627}a^{5}+\frac{7821}{1627}a^{4}+\frac{209665}{1627}a^{3}-\frac{40351}{1627}a^{2}-\frac{78510}{1627}a+\frac{20604}{1627}$, $\frac{14452}{1627}a^{11}-\frac{31918}{1627}a^{10}+\frac{3675}{1627}a^{9}+\frac{2916}{1627}a^{8}-\frac{99084}{1627}a^{7}+\frac{8489}{1627}a^{6}-\frac{152640}{1627}a^{5}-\frac{5114}{1627}a^{4}+\frac{298543}{1627}a^{3}-\frac{24776}{1627}a^{2}-\frac{104829}{1627}a+\frac{19372}{1627}$, $\frac{3341}{1627}a^{11}-\frac{7805}{1627}a^{10}+\frac{2023}{1627}a^{9}-\frac{59}{1627}a^{8}-\frac{22273}{1627}a^{7}+\frac{3672}{1627}a^{6}-\frac{35819}{1627}a^{5}+\frac{2602}{1627}a^{4}+\frac{65788}{1627}a^{3}-\frac{13307}{1627}a^{2}-\frac{23762}{1627}a+\frac{8792}{1627}$, $\frac{771}{1627}a^{11}-\frac{1676}{1627}a^{10}+\frac{592}{1627}a^{9}-\frac{1140}{1627}a^{8}-\frac{4389}{1627}a^{7}-\frac{279}{1627}a^{6}-\frac{10769}{1627}a^{5}+\frac{1852}{1627}a^{4}+\frac{10426}{1627}a^{3}+\frac{4939}{1627}a^{2}+\frac{2276}{1627}a-\frac{1976}{1627}$
| 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: | \( 164.286322613 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 164.286322613 \cdot 1}{2\cdot\sqrt{40634924000000}}\cr\approx \mathstrut & 0.204569680064 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*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 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*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 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*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 - 3*x^11 + 2*x^10 - 7*x^8 + 6*x^7 - 11*x^6 + 8*x^5 + 21*x^4 - 18*x^3 - 6*x^2 + 7*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
$A_4^2:D_4$ (as 12T208):
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 1152 |
| The 44 conjugacy class representatives for $A_4^2:D_4$ |
| Character table for $A_4^2:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.1922000.1 |
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 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | R | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(31\)
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 31.6.4.2 | $x^{6} - 899 x^{3} + 2883$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |