Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*x - 1)
gp: K = bnfinit(y^12 - 4*y^11 - y^10 + 20*y^9 - 20*y^8 - 16*y^7 + 41*y^6 - 8*y^5 - 30*y^4 + 10*y^3 + 9*y^2 - 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*x - 1)
\( x^{12} - 4x^{11} - x^{10} + 20x^{9} - 20x^{8} - 16x^{7} + 41x^{6} - 8x^{5} - 30x^{4} + 10x^{3} + 9x^{2} - 2x - 1 \)
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: | $[10, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-492092096000000\)
\(\medspace = -\,2^{12}\cdot 5^{6}\cdot 19^{4}\cdot 59\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.76\) | 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: | $2^{31/24}5^{1/2}19^{2/3}59^{1/2}\approx 299.39450122576613$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\), \(59\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-59}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{49}a^{11}-\frac{16}{49}a^{10}-\frac{5}{49}a^{9}-\frac{18}{49}a^{8}-\frac{16}{49}a^{6}-\frac{12}{49}a^{5}-\frac{11}{49}a^{4}+\frac{4}{49}a^{3}+\frac{11}{49}a^{2}+\frac{24}{49}a+\frac{4}{49}$
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: | $10$ | 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: |
$\frac{120}{49}a^{11}-\frac{352}{49}a^{10}-\frac{600}{49}a^{9}+\frac{2201}{49}a^{8}-2a^{7}-\frac{3880}{49}a^{6}+\frac{3264}{49}a^{5}+\frac{2698}{49}a^{4}-\frac{3832}{49}a^{3}-\frac{1228}{49}a^{2}+\frac{1018}{49}a+\frac{235}{49}$, $\frac{128}{49}a^{11}-\frac{480}{49}a^{10}-\frac{199}{49}a^{9}+\frac{2302}{49}a^{8}-40a^{7}-\frac{1656}{49}a^{6}+\frac{3658}{49}a^{5}-\frac{232}{49}a^{4}-\frac{2428}{49}a^{3}-\frac{62}{49}a^{2}+\frac{475}{49}a+\frac{120}{49}$, $\frac{57}{49}a^{11}-\frac{324}{49}a^{10}+\frac{352}{49}a^{9}+\frac{1081}{49}a^{8}-58a^{7}+\frac{1440}{49}a^{6}+\frac{2403}{49}a^{5}-\frac{3322}{49}a^{4}-\frac{66}{49}a^{3}+\frac{1656}{49}a^{2}-\frac{102}{49}a-\frac{213}{49}$, $\frac{60}{49}a^{11}-\frac{176}{49}a^{10}-\frac{300}{49}a^{9}+\frac{1076}{49}a^{8}+a^{7}-\frac{1940}{49}a^{6}+\frac{1240}{49}a^{5}+\frac{1839}{49}a^{4}-\frac{1916}{49}a^{3}-\frac{1104}{49}a^{2}+\frac{705}{49}a+\frac{289}{49}$, $\frac{8}{49}a^{11}-\frac{128}{49}a^{10}+\frac{401}{49}a^{9}+\frac{101}{49}a^{8}-38a^{7}+\frac{2224}{49}a^{6}+\frac{394}{49}a^{5}-\frac{2930}{49}a^{4}+\frac{1404}{49}a^{3}+\frac{1166}{49}a^{2}-\frac{543}{49}a-\frac{115}{49}$, $\frac{156}{49}a^{11}-\frac{536}{49}a^{10}-\frac{437}{49}a^{9}+\frac{2778}{49}a^{8}-31a^{7}-\frac{2986}{49}a^{6}+\frac{4106}{49}a^{5}+\frac{1224}{49}a^{4}-\frac{3394}{49}a^{3}-\frac{930}{49}a^{2}+\frac{853}{49}a+\frac{330}{49}$, $a^{11}-4a^{10}+16a^{8}-20a^{7}+21a^{5}-8a^{4}-9a^{3}+2a^{2}$, $\frac{156}{49}a^{11}-\frac{634}{49}a^{10}-\frac{94}{49}a^{9}+\frac{3023}{49}a^{8}-67a^{7}-\frac{1810}{49}a^{6}+\frac{5870}{49}a^{5}-\frac{1814}{49}a^{4}-\frac{3639}{49}a^{3}+\frac{1373}{49}a^{2}+\frac{706}{49}a-\frac{209}{49}$, $\frac{71}{49}a^{11}-\frac{156}{49}a^{10}-\frac{551}{49}a^{9}+\frac{1221}{49}a^{8}+18a^{7}-\frac{3096}{49}a^{6}+\frac{1255}{49}a^{5}+\frac{3090}{49}a^{4}-\frac{2362}{49}a^{3}-\frac{1718}{49}a^{2}+\frac{626}{49}a+\frac{333}{49}$, $\frac{349}{49}a^{11}-\frac{1272}{49}a^{10}-\frac{716}{49}a^{9}+\frac{6409}{49}a^{8}-99a^{7}-\frac{5731}{49}a^{6}+\frac{10953}{49}a^{5}-\frac{213}{49}a^{4}-\frac{7718}{49}a^{3}+\frac{458}{49}a^{2}+\frac{1369}{49}a+\frac{171}{49}$
| 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: | \( 1352.48458763 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{1}\cdot 1352.48458763 \cdot 1}{2\cdot\sqrt{492092096000000}}\cr\approx \mathstrut & 0.196136743065 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*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 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*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 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*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 - x^10 + 20*x^9 - 20*x^8 - 16*x^7 + 41*x^6 - 8*x^5 - 30*x^4 + 10*x^3 + 9*x^2 - 2*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 $A_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.722000.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.29 | $x^{12} + 4 x^{8} + 4 x^{7} - 2 x^{6} + 4 x^{4} + 8 x^{3} - 4 x + 4$ | $6$ | $2$ | $12$ | 12T159 | $[4/3, 4/3, 4/3, 4/3]_{3}^{12}$ |
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 19.6.4.2 | $x^{6} - 342 x^{3} + 722$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(59\)
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.3.0.1 | $x^{3} + 5 x + 57$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 59.3.0.1 | $x^{3} + 5 x + 57$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |