Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98)
gp: K = bnfinit(y^12 - y^11 + 4*y^10 - 3*y^9 + 59*y^8 + 8*y^7 + 207*y^6 + 103*y^5 + 406*y^4 + 143*y^3 + 489*y^2 - 70*y + 98, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98)
\( x^{12} - x^{11} + 4 x^{10} - 3 x^{9} + 59 x^{8} + 8 x^{7} + 207 x^{6} + 103 x^{5} + 406 x^{4} + \cdots + 98 \)
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: |
\(4492085992834072576\)
\(\medspace = 2^{17}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.84\) | 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: | $2^{17/6}17^{11/12}\approx 95.68190916377696$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{43559100296012}a^{11}+\frac{3437934846489}{21779550148006}a^{10}-\frac{4659069315606}{10889775074003}a^{9}-\frac{4513477328657}{43559100296012}a^{8}-\frac{2873341398657}{21779550148006}a^{7}-\frac{3048734560777}{21779550148006}a^{6}+\frac{20706987925229}{43559100296012}a^{5}+\frac{3389465756011}{21779550148006}a^{4}+\frac{591861284519}{3111364306858}a^{3}-\frac{14328988193937}{43559100296012}a^{2}-\frac{4615147736565}{10889775074003}a+\frac{537641291955}{3111364306858}$
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
$C_{2}\times C_{2}$, which has order $4$ (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: | $5$ | 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: |
$\frac{222587916765}{21779550148006}a^{11}-\frac{664116471197}{21779550148006}a^{10}+\frac{1633349347205}{21779550148006}a^{9}-\frac{1536181835012}{10889775074003}a^{8}+\frac{7922587387658}{10889775074003}a^{7}-\frac{12624922583271}{10889775074003}a^{6}+\frac{58232536193253}{21779550148006}a^{5}-\frac{79041141624887}{21779550148006}a^{4}+\frac{12373257356287}{3111364306858}a^{3}-\frac{59909599139405}{10889775074003}a^{2}+\frac{18011660840759}{10889775074003}a-\frac{7626566977863}{1555682153429}$, $\frac{354210820639}{43559100296012}a^{11}-\frac{80931219833}{21779550148006}a^{10}+\frac{317181868224}{10889775074003}a^{9}-\frac{1298844999219}{43559100296012}a^{8}+\frac{10581920338177}{21779550148006}a^{7}+\frac{7220042192491}{21779550148006}a^{6}+\frac{72305019531859}{43559100296012}a^{5}+\frac{16450487423121}{21779550148006}a^{4}+\frac{9924894986989}{3111364306858}a^{3}+\frac{147093834189405}{43559100296012}a^{2}-\frac{10972677805377}{10889775074003}a+\frac{2832466893201}{3111364306858}$, $\frac{172931056143}{43559100296012}a^{11}-\frac{18196160601}{21779550148006}a^{10}-\frac{3268444642}{10889775074003}a^{9}+\frac{1253669317533}{43559100296012}a^{8}+\frac{3033151374619}{21779550148006}a^{7}+\frac{6996050906701}{21779550148006}a^{6}+\frac{453613825115}{43559100296012}a^{5}+\frac{38039316403593}{21779550148006}a^{4}-\frac{4140979466781}{3111364306858}a^{3}+\frac{120615768949449}{43559100296012}a^{2}-\frac{15532430051087}{10889775074003}a+\frac{1777165189201}{3111364306858}$, $\frac{292495090913}{21779550148006}a^{11}-\frac{446054981229}{21779550148006}a^{10}+\frac{794266452681}{21779550148006}a^{9}-\frac{371620795517}{10889775074003}a^{8}+\frac{8264210927979}{10889775074003}a^{7}-\frac{2916000586784}{10889775074003}a^{6}+\frac{29673974145545}{21779550148006}a^{5}+\frac{4501885778599}{21779550148006}a^{4}+\frac{7617035827601}{3111364306858}a^{3}-\frac{23259997834956}{10889775074003}a^{2}+\frac{10006760899805}{10889775074003}a-\frac{2637310962877}{1555682153429}$, $\frac{356799526477}{21779550148006}a^{11}-\frac{1002586845563}{21779550148006}a^{10}+\frac{2226192292073}{21779550148006}a^{9}-\frac{1966615411792}{10889775074003}a^{8}+\frac{11779911911666}{10889775074003}a^{7}-\frac{17683664257755}{10889775074003}a^{6}+\frac{75971366604813}{21779550148006}a^{5}-\frac{99407069828693}{21779550148006}a^{4}+\frac{13687825257695}{3111364306858}a^{3}-\frac{91300918972519}{10889775074003}a^{2}+\frac{25723440526615}{10889775074003}a-\frac{16390816530933}{1555682153429}$
| 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: | \( 16124.0322786 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 16124.0322786 \cdot 4}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 0.936178868516 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98)
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 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98, 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 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98);
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 - x^11 + 4*x^10 - 3*x^9 + 59*x^8 + 8*x^7 + 207*x^6 + 103*x^5 + 406*x^4 + 143*x^3 + 489*x^2 - 70*x + 98);
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
$S_4^2:D_4$ (as 12T260):
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 4608 |
| The 65 conjugacy class representatives for $S_4^2:D_4$ |
| Character table for $S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 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 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: | 12.2.561510749104259072.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.10.6 | $x^{4} + 4 x^{3} + 12 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |