Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64)
gp: K = bnfinit(y^12 - 3*y^11 + 10*y^10 - 11*y^9 + 27*y^8 - 36*y^7 + 31*y^6 - 9*y^5 + 10*y^4 + 107*y^3 - 39*y^2 - 8*y + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64)
\( x^{12} - 3 x^{11} + 10 x^{10} - 11 x^{9} + 27 x^{8} - 36 x^{7} + 31 x^{6} - 9 x^{5} + 10 x^{4} + \cdots + 64 \)
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: |
\(4469547301936929\)
\(\medspace = 3^{6}\cdot 19^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.15\) | 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: | $3^{1/2}19^{5/6}\approx 20.14599065465418$ | ||
| Ramified primes: |
\(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{12}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{1}{6}a^{2}+\frac{1}{12}a$, $\frac{1}{24}a^{8}-\frac{1}{24}a^{7}-\frac{1}{12}a^{6}-\frac{1}{4}a^{4}-\frac{5}{12}a^{3}+\frac{7}{24}a^{2}+\frac{1}{8}a+\frac{1}{3}$, $\frac{1}{24}a^{9}-\frac{1}{24}a^{7}-\frac{1}{12}a^{6}-\frac{1}{12}a^{5}+\frac{5}{24}a^{3}-\frac{5}{12}a^{2}+\frac{1}{24}a+\frac{1}{3}$, $\frac{1}{360}a^{10}-\frac{7}{360}a^{9}+\frac{1}{360}a^{8}-\frac{11}{360}a^{7}+\frac{1}{90}a^{6}-\frac{5}{36}a^{5}-\frac{7}{360}a^{4}+\frac{11}{40}a^{3}+\frac{53}{360}a^{2}-\frac{1}{120}a+\frac{14}{45}$, $\frac{1}{16560}a^{11}+\frac{7}{8280}a^{10}+\frac{4}{1035}a^{9}+\frac{59}{3312}a^{8}+\frac{7}{4140}a^{7}+\frac{271}{4140}a^{6}-\frac{1597}{16560}a^{5}+\frac{457}{2760}a^{4}-\frac{1477}{4140}a^{3}+\frac{49}{368}a^{2}-\frac{247}{2070}a-\frac{57}{115}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$
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: |
\( -\frac{7}{5520} a^{11} - \frac{13}{1380} a^{10} + \frac{5}{184} a^{9} - \frac{213}{1840} a^{8} + \frac{113}{920} a^{7} - \frac{157}{460} a^{6} + \frac{3359}{5520} a^{5} - \frac{739}{1380} a^{4} + \frac{89}{184} a^{3} - \frac{3337}{5520} a^{2} - \frac{2123}{2760} a + \frac{116}{115} \)
(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: |
$\frac{1}{360}a^{11}-\frac{1}{36}a^{10}+\frac{11}{180}a^{9}-\frac{59}{360}a^{8}+\frac{11}{180}a^{7}-\frac{19}{45}a^{6}+\frac{23}{360}a^{5}+\frac{1}{4}a^{4}-\frac{107}{180}a^{3}+\frac{161}{120}a^{2}-\frac{487}{180}a+\frac{1}{15}$, $\frac{1}{240}a^{11}-\frac{13}{360}a^{10}+\frac{17}{180}a^{9}-\frac{173}{720}a^{8}+\frac{71}{360}a^{7}-\frac{11}{18}a^{6}+\frac{289}{720}a^{5}-\frac{119}{360}a^{4}+\frac{7}{60}a^{3}+\frac{641}{720}a^{2}-\frac{289}{120}a-\frac{19}{9}$, $\frac{17}{5520}a^{11}-\frac{43}{2070}a^{10}+\frac{101}{1656}a^{9}-\frac{2573}{16560}a^{8}+\frac{1703}{8280}a^{7}-\frac{2417}{4140}a^{6}+\frac{10093}{16560}a^{5}-\frac{1147}{1035}a^{4}+\frac{403}{552}a^{3}-\frac{15379}{16560}a^{2}+\frac{1903}{2760}a-\frac{449}{1035}$, $\frac{169}{3312}a^{11}-\frac{133}{828}a^{10}+\frac{427}{828}a^{9}-\frac{2033}{3312}a^{8}+\frac{2297}{1656}a^{7}-\frac{1811}{828}a^{6}+\frac{6107}{3312}a^{5}-\frac{24}{23}a^{4}+\frac{601}{414}a^{3}+\frac{4247}{1104}a^{2}-\frac{3235}{1656}a-\frac{103}{69}$, $\frac{571}{16560}a^{11}-\frac{26}{207}a^{10}+\frac{1753}{4140}a^{9}-\frac{10679}{16560}a^{8}+\frac{10961}{8280}a^{7}-\frac{8513}{4140}a^{6}+\frac{37553}{16560}a^{5}-\frac{445}{276}a^{4}+\frac{2287}{2070}a^{3}+\frac{4747}{1840}a^{2}-\frac{33607}{8280}a+\frac{293}{345}$
| 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: | \( 2885.43572816 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2885.43572816 \cdot 3}{6\cdot\sqrt{4469547301936929}}\cr\approx \mathstrut & 1.32778834009 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64)
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 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64, 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 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64);
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 + 10*x^10 - 11*x^9 + 27*x^8 - 36*x^7 + 31*x^6 - 9*x^5 + 10*x^4 + 107*x^3 - 39*x^2 - 8*x + 64);
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
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 12 |
| The 6 conjugacy class representatives for $D_6$ |
| Character table for $D_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), 3.1.1083.1 x3, \(\Q(\sqrt{-3}, \sqrt{-19})\), 6.0.3518667.2, 6.0.22284891.1 x3, 6.2.66854673.1 x3 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(19\)
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |