Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 1)
gp: K = bnfinit(y^18 - 2*y^17 - y^16 + 6*y^15 - 4*y^14 - 10*y^13 + 11*y^12 + 9*y^11 - 16*y^10 - 5*y^9 + 16*y^8 + 5*y^7 - 15*y^6 - 5*y^5 + 12*y^4 + y^3 - 5*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 1)
\( x^{18} - 2 x^{17} - x^{16} + 6 x^{15} - 4 x^{14} - 10 x^{13} + 11 x^{12} + 9 x^{11} - 16 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6425148232879087321\)
\(\medspace = 23^{6}\cdot 208333^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.09\) | 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: | $23^{1/2}208333^{1/2}\approx 2188.985838236511$ | ||
| Ramified primes: |
\(23\), \(208333\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{59}a^{16}-\frac{23}{59}a^{15}+\frac{27}{59}a^{14}-\frac{8}{59}a^{13}-\frac{26}{59}a^{12}-\frac{13}{59}a^{11}+\frac{19}{59}a^{10}-\frac{21}{59}a^{9}-\frac{19}{59}a^{8}-\frac{22}{59}a^{7}-\frac{22}{59}a^{6}-\frac{25}{59}a^{5}+\frac{18}{59}a^{4}+\frac{18}{59}a^{3}-\frac{1}{59}a^{2}-\frac{26}{59}a-\frac{7}{59}$, $\frac{1}{2537}a^{17}+\frac{13}{2537}a^{16}-\frac{1096}{2537}a^{15}+\frac{551}{2537}a^{14}-\frac{1199}{2537}a^{13}-\frac{64}{2537}a^{12}-\frac{390}{2537}a^{11}+\frac{781}{2537}a^{10}-\frac{126}{2537}a^{9}-\frac{175}{2537}a^{8}-\frac{932}{2537}a^{7}-\frac{19}{59}a^{6}+\frac{888}{2537}a^{5}-\frac{101}{2537}a^{4}+\frac{647}{2537}a^{3}-\frac{829}{2537}a^{2}-\frac{1002}{2537}a+\frac{751}{2537}$
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: | $9$ | 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: |
$\frac{100}{59}a^{17}-\frac{152}{59}a^{16}-\frac{153}{59}a^{15}+\frac{497}{59}a^{14}-\frac{196}{59}a^{13}-\frac{980}{59}a^{12}+\frac{585}{59}a^{11}+\frac{952}{59}a^{10}-\frac{988}{59}a^{9}-\frac{768}{59}a^{8}+\frac{930}{59}a^{7}+\frac{807}{59}a^{6}-\frac{865}{59}a^{5}-\frac{777}{59}a^{4}+\frac{627}{59}a^{3}+\frac{267}{59}a^{2}-\frac{203}{59}a-\frac{50}{59}$, $a$, $\frac{2459}{2537}a^{17}-\frac{3766}{2537}a^{16}-\frac{3436}{2537}a^{15}+\frac{12105}{2537}a^{14}-\frac{6238}{2537}a^{13}-\frac{22399}{2537}a^{12}+\frac{15456}{2537}a^{11}+\frac{18718}{2537}a^{10}-\frac{23712}{2537}a^{9}-\frac{12709}{2537}a^{8}+\frac{21612}{2537}a^{7}+\frac{353}{59}a^{6}-\frac{18223}{2537}a^{5}-\frac{16288}{2537}a^{4}+\frac{14163}{2537}a^{3}+\frac{1452}{2537}a^{2}-\frac{5049}{2537}a+\frac{1278}{2537}$, $\frac{1620}{2537}a^{17}-\frac{1558}{2537}a^{16}-\frac{4565}{2537}a^{15}+\frac{7938}{2537}a^{14}+\frac{1779}{2537}a^{13}-\frac{20475}{2537}a^{12}+\frac{2192}{2537}a^{11}+\frac{26175}{2537}a^{10}-\frac{10749}{2537}a^{9}-\frac{26274}{2537}a^{8}+\frac{12704}{2537}a^{7}+\frac{616}{59}a^{6}-\frac{10368}{2537}a^{5}-\frac{27826}{2537}a^{4}+\frac{6766}{2537}a^{3}+\frac{14100}{2537}a^{2}-\frac{2613}{2537}a-\frac{5182}{2537}$, $\frac{4769}{2537}a^{17}-\frac{5857}{2537}a^{16}-\frac{7828}{2537}a^{15}+\frac{19339}{2537}a^{14}-\frac{4793}{2537}a^{13}-\frac{42916}{2537}a^{12}+\frac{11622}{2537}a^{11}+\frac{40435}{2537}a^{10}-\frac{28392}{2537}a^{9}-\frac{37527}{2537}a^{8}+\frac{26518}{2537}a^{7}+\frac{981}{59}a^{6}-\frac{20580}{2537}a^{5}-\frac{33695}{2537}a^{4}+\frac{12161}{2537}a^{3}+\frac{8648}{2537}a^{2}-\frac{2915}{2537}a-\frac{2703}{2537}$, $\frac{961}{2537}a^{17}-\frac{3933}{2537}a^{16}+\frac{4458}{2537}a^{15}+\frac{4825}{2537}a^{14}-\frac{16179}{2537}a^{13}+\frac{5318}{2537}a^{12}+\frac{23949}{2537}a^{11}-\frac{23287}{2537}a^{10}-\frac{19692}{2537}a^{9}+\frac{32291}{2537}a^{8}+\frac{8638}{2537}a^{7}-\frac{651}{59}a^{6}-\frac{9556}{2537}a^{5}+\frac{28413}{2537}a^{4}+\frac{8974}{2537}a^{3}-\frac{24217}{2537}a^{2}+\frac{1998}{2537}a+\frac{7094}{2537}$, $\frac{1737}{2537}a^{17}-\frac{2273}{2537}a^{16}-\frac{2722}{2537}a^{15}+\frac{6959}{2537}a^{14}-\frac{1377}{2537}a^{13}-\frac{15493}{2537}a^{12}+\frac{3389}{2537}a^{11}+\frac{16717}{2537}a^{10}-\frac{8979}{2537}a^{9}-\frac{16950}{2537}a^{8}+\frac{8666}{2537}a^{7}+\frac{422}{59}a^{6}-\frac{7866}{2537}a^{5}-\frac{16466}{2537}a^{4}+\frac{1625}{2537}a^{3}+\frac{5601}{2537}a^{2}+\frac{1714}{2537}a-\frac{3143}{2537}$, $\frac{3860}{2537}a^{17}-\frac{7784}{2537}a^{16}-\frac{2671}{2537}a^{15}+\frac{21451}{2537}a^{14}-\frac{16433}{2537}a^{13}-\frac{33846}{2537}a^{12}+\frac{39676}{2537}a^{11}+\frac{25816}{2537}a^{10}-\frac{55586}{2537}a^{9}-\frac{13085}{2537}a^{8}+\frac{49783}{2537}a^{7}+\frac{330}{59}a^{6}-\frac{47537}{2537}a^{5}-\frac{15029}{2537}a^{4}+\frac{35885}{2537}a^{3}+\frac{1367}{2537}a^{2}-\frac{11394}{2537}a-\frac{1103}{2537}$, $\frac{1574}{2537}a^{17}+\frac{123}{2537}a^{16}-\frac{6566}{2537}a^{15}+\frac{3533}{2537}a^{14}+\frac{10794}{2537}a^{13}-\frac{20971}{2537}a^{12}-\frac{17106}{2537}a^{11}+\frac{33550}{2537}a^{10}+\frac{13150}{2537}a^{9}-\frac{38692}{2537}a^{8}-\frac{4710}{2537}a^{7}+\frac{973}{59}a^{6}+\frac{8511}{2537}a^{5}-\frac{37972}{2537}a^{4}-\frac{12418}{2537}a^{3}+\frac{19511}{2537}a^{2}+\frac{4521}{2537}a-\frac{4941}{2537}$
| 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: | \( 79.1741198129 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 79.1741198129 \cdot 1}{2\cdot\sqrt{6425148232879087321}}\cr\approx \mathstrut & 0.151743679505 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 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^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 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^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 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^18 - 2*x^17 - x^16 + 6*x^15 - 4*x^14 - 10*x^13 + 11*x^12 + 9*x^11 - 16*x^10 - 5*x^9 + 16*x^8 + 5*x^7 - 15*x^6 - 5*x^5 + 12*x^4 + x^3 - 5*x^2 + 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
$A_4^3.(C_2\times S_4)$ (as 18T776):
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 82944 |
| The 65 conjugacy class representatives for $A_4^3.(C_2\times S_4)$ |
| Character table for $A_4^3.(C_2\times S_4)$ |
Intermediate fields
| 3.1.23.1, 9.3.2534787611.1 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.58300115053.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{7}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(208333\)
| $\Q_{208333}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{208333}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |