Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16)
gp: K = bnfinit(y^12 - 5*y^11 - 2*y^10 + 50*y^9 - 83*y^8 - 75*y^7 + 380*y^6 - 330*y^5 - 100*y^4 + 356*y^3 - 240*y^2 + 80*y - 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16)
\( x^{12} - 5 x^{11} - 2 x^{10} + 50 x^{9} - 83 x^{8} - 75 x^{7} + 380 x^{6} - 330 x^{5} - 100 x^{4} + \cdots - 16 \)
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: |
\(-280755374552129536\)
\(\medspace = -\,2^{13}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.45\) | 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: | $2^{11/6}17^{11/12}\approx 47.84095458188848$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-34}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{1831448}a^{11}+\frac{190315}{1831448}a^{10}+\frac{101851}{915724}a^{9}+\frac{3891}{457862}a^{8}-\frac{226743}{1831448}a^{7}-\frac{234335}{1831448}a^{6}-\frac{16356}{228931}a^{5}-\frac{83843}{457862}a^{4}+\frac{88854}{228931}a^{3}+\frac{36433}{457862}a^{2}+\frac{33186}{228931}a-\frac{17829}{228931}$
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
$C_{2}$, which has order $2$ (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: | $8$ | 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: |
$\frac{59533}{1831448}a^{11}-\frac{230157}{1831448}a^{10}-\frac{99907}{457862}a^{9}+\frac{325762}{228931}a^{8}-\frac{1834983}{1831448}a^{7}-\frac{7851931}{1831448}a^{6}+\frac{7699765}{915724}a^{5}+\frac{207568}{228931}a^{4}-\frac{1994883}{228931}a^{3}+\frac{2362805}{457862}a^{2}+\frac{216539}{228931}a-\frac{89741}{228931}$, $\frac{58959}{1831448}a^{11}-\frac{41949}{1831448}a^{10}-\frac{256907}{457862}a^{9}+\frac{124769}{228931}a^{8}+\frac{5608483}{1831448}a^{7}-\frac{9328655}{1831448}a^{6}-\frac{5111679}{915724}a^{5}+\frac{4124581}{228931}a^{4}+\frac{801706}{228931}a^{3}-\frac{4356015}{457862}a^{2}+\frac{626910}{228931}a-\frac{386721}{228931}$, $\frac{135025}{1831448}a^{11}-\frac{679737}{1831448}a^{10}-\frac{35172}{228931}a^{9}+\frac{3405225}{915724}a^{8}-\frac{10646047}{1831448}a^{7}-\frac{11976415}{1831448}a^{6}+\frac{25067307}{915724}a^{5}-\frac{16780351}{915724}a^{4}-\frac{3280601}{228931}a^{3}+\frac{9711599}{457862}a^{2}-\frac{1758261}{228931}a+\frac{535533}{228931}$, $\frac{316235}{1831448}a^{11}-\frac{1238013}{1831448}a^{10}-\frac{526961}{457862}a^{9}+\frac{7029381}{915724}a^{8}-\frac{10209197}{1831448}a^{7}-\frac{40713743}{1831448}a^{6}+\frac{40572341}{915724}a^{5}-\frac{207887}{915724}a^{4}-\frac{8487766}{228931}a^{3}+\frac{11196937}{457862}a^{2}+\frac{148739}{228931}a+\frac{416715}{228931}$, $\frac{171201}{1831448}a^{11}-\frac{715191}{1831448}a^{10}-\frac{113072}{228931}a^{9}+\frac{3799589}{915724}a^{8}-\frac{7498051}{1831448}a^{7}-\frac{18374513}{1831448}a^{6}+\frac{23145913}{915724}a^{5}-\frac{7160347}{915724}a^{4}-\frac{6865867}{457862}a^{3}+\frac{4191227}{228931}a^{2}-\frac{1735689}{228931}a+\frac{223325}{228931}$, $\frac{14973}{1831448}a^{11}-\frac{146593}{1831448}a^{10}+\frac{26408}{228931}a^{9}+\frac{170200}{228931}a^{8}-\frac{4096967}{1831448}a^{7}-\frac{559311}{1831448}a^{6}+\frac{7785851}{915724}a^{5}-\frac{2136662}{228931}a^{4}-\frac{1280685}{228931}a^{3}+\frac{4318425}{457862}a^{2}-\frac{573085}{228931}a-\frac{20071}{228931}$, $\frac{143235}{1831448}a^{11}-\frac{876593}{1831448}a^{10}+\frac{228941}{915724}a^{9}+\frac{1940779}{457862}a^{8}-\frac{19696425}{1831448}a^{7}-\frac{484091}{1831448}a^{6}+\frac{8827572}{228931}a^{5}-\frac{12700524}{228931}a^{4}+\frac{1873055}{228931}a^{3}+\frac{20373469}{457862}a^{2}-\frac{9283814}{228931}a+\frac{2506731}{228931}$, $\frac{223309}{457862}a^{11}-\frac{1915251}{915724}a^{10}-\frac{2259943}{915724}a^{9}+\frac{20734389}{915724}a^{8}-\frac{11156281}{457862}a^{7}-\frac{49353235}{915724}a^{6}+\frac{134526803}{915724}a^{5}-\frac{51996001}{915724}a^{4}-\frac{40140893}{457862}a^{3}+\frac{25087317}{228931}a^{2}-\frac{9357879}{228931}a+\frac{2827543}{228931}$
| 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: | \( 69919.1497116 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 69919.1497116 \cdot 2}{2\cdot\sqrt{280755374552129536}}\cr\approx \mathstrut & 2.09484395480 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16)
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 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16, 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 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16);
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 - 5*x^11 - 2*x^10 + 50*x^9 - 83*x^8 - 75*x^7 + 380*x^6 - 330*x^5 - 100*x^4 + 356*x^3 - 240*x^2 + 80*x - 16);
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
$S_3^2:C_4$ (as 12T80):
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 144 |
| The 18 conjugacy class representatives for $S_3^2:C_4$ |
| Character table for $S_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.39304.1, 6.4.45435424.1 |
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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.8.35094421819016192.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |