Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81)
gp: K = bnfinit(y^12 - 4*y^11 + 13*y^10 - 22*y^9 + 42*y^8 - 2*y^7 - 3*y^6 - 4*y^5 - 308*y^4 - 6*y^3 - 234*y^2 - 162*y + 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81)
\( x^{12} - 4 x^{11} + 13 x^{10} - 22 x^{9} + 42 x^{8} - 2 x^{7} - 3 x^{6} - 4 x^{5} - 308 x^{4} + \cdots + 81 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1123021498208518144\)
\(\medspace = 2^{15}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.93\) | 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: | $2^{19/8}17^{11/12}\approx 69.63983780733449$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{36}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{9}a^{7}-\frac{7}{18}a^{5}-\frac{1}{12}a^{4}-\frac{1}{9}a^{3}-\frac{11}{36}a^{2}-\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{38347709688}a^{11}+\frac{26464553}{38347709688}a^{10}+\frac{1442088149}{19173854844}a^{9}-\frac{366458840}{4793463711}a^{8}-\frac{968109439}{6391284948}a^{7}+\frac{1563255875}{4793463711}a^{6}+\frac{1117939729}{4260856632}a^{5}-\frac{15469469203}{38347709688}a^{4}-\frac{10613264279}{38347709688}a^{3}+\frac{2629558853}{12782569896}a^{2}-\frac{300678477}{1420285544}a-\frac{368286099}{1420285544}$
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
$C_{2}$, which has order $2$
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: | $7$ | 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: |
$\frac{92444383}{9586927422}a^{11}-\frac{452720119}{9586927422}a^{10}+\frac{796455929}{4793463711}a^{9}-\frac{3468519427}{9586927422}a^{8}+\frac{1187154346}{1597821237}a^{7}-\frac{7370584355}{9586927422}a^{6}+\frac{2725786325}{3195642474}a^{5}-\frac{6125902121}{4793463711}a^{4}-\frac{12509016977}{9586927422}a^{3}+\frac{2441329327}{3195642474}a^{2}-\frac{1482785201}{1065214158}a+\frac{65916351}{177535693}$, $\frac{224821669}{38347709688}a^{11}-\frac{900203785}{38347709688}a^{10}+\frac{1473767225}{19173854844}a^{9}-\frac{1351673743}{9586927422}a^{8}+\frac{1809678955}{6391284948}a^{7}-\frac{440569990}{4793463711}a^{6}+\frac{388903655}{12782569896}a^{5}-\frac{4581498085}{38347709688}a^{4}-\frac{84772299203}{38347709688}a^{3}+\frac{4563240655}{12782569896}a^{2}-\frac{7076281511}{4260856632}a+\frac{889208747}{1420285544}$, $\frac{13078904}{4793463711}a^{11}-\frac{46465457}{19173854844}a^{10}-\frac{25739947}{4793463711}a^{9}+\frac{415247527}{4793463711}a^{8}-\frac{217868915}{1065214158}a^{7}+\frac{6451631575}{9586927422}a^{6}-\frac{1179778891}{1597821237}a^{5}+\frac{17812685449}{19173854844}a^{4}-\frac{10153798591}{4793463711}a^{3}-\frac{569683383}{710142772}a^{2}-\frac{356765101}{355071386}a-\frac{1742781707}{710142772}$, $\frac{411459545}{38347709688}a^{11}-\frac{1272582083}{38347709688}a^{10}+\frac{2133968689}{19173854844}a^{9}-\frac{791172610}{4793463711}a^{8}+\frac{2621544637}{6391284948}a^{7}+\frac{252608872}{4793463711}a^{6}+\frac{693077155}{1420285544}a^{5}-\frac{14362445447}{38347709688}a^{4}-\frac{146473080823}{38347709688}a^{3}-\frac{43146920951}{12782569896}a^{2}-\frac{7674529997}{1420285544}a-\frac{527123767}{1420285544}$, $\frac{124810727}{38347709688}a^{11}-\frac{700461005}{38347709688}a^{10}+\frac{1472231491}{19173854844}a^{9}-\frac{2036325473}{9586927422}a^{8}+\frac{3208975111}{6391284948}a^{7}-\frac{6860285215}{9586927422}a^{6}+\frac{3090926047}{4260856632}a^{5}-\frac{2931281669}{38347709688}a^{4}-\frac{76504471153}{38347709688}a^{3}+\frac{32109972199}{12782569896}a^{2}-\frac{4000759371}{1420285544}a+\frac{1672982547}{1420285544}$, $\frac{191260205}{38347709688}a^{11}-\frac{630264923}{38347709688}a^{10}+\frac{982365673}{19173854844}a^{9}-\frac{270560539}{4793463711}a^{8}+\frac{770637001}{6391284948}a^{7}+\frac{754783342}{4793463711}a^{6}+\frac{265335133}{4260856632}a^{5}+\frac{3868656097}{38347709688}a^{4}-\frac{44740383979}{38347709688}a^{3}-\frac{12926660375}{12782569896}a^{2}-\frac{1040174897}{1420285544}a-\frac{2657211055}{1420285544}$, $\frac{839894087}{38347709688}a^{11}-\frac{2799878639}{38347709688}a^{10}+\frac{4705206271}{19173854844}a^{9}-\frac{3836767679}{9586927422}a^{8}+\frac{678984273}{710142772}a^{7}-\frac{887669978}{4793463711}a^{6}+\frac{15078744125}{12782569896}a^{5}-\frac{44778172787}{38347709688}a^{4}-\frac{278744767657}{38347709688}a^{3}-\frac{7138241751}{1420285544}a^{2}-\frac{41071429325}{4260856632}a+\frac{3391897597}{1420285544}$
| 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: | \( 37665.0870587 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 37665.0870587 \cdot 2}{2\cdot\sqrt{1123021498208518144}}\cr\approx \mathstrut & 0.886307443408 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81)
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 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81, 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 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81);
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 + 13*x^10 - 22*x^9 + 42*x^8 - 2*x^7 - 3*x^6 - 4*x^5 - 308*x^4 - 6*x^3 - 234*x^2 - 162*x + 81);
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
$S_4^2:C_4$ (as 12T238):
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 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_4$ |
| Character table for $S_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 6.2.11358856.1 |
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 |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.561510749104259072.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.6.6.6 | $x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |