Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 1)
gp: K = bnfinit(y^18 - 5*y^17 + 14*y^16 - 27*y^15 + 44*y^14 - 68*y^13 + 98*y^12 - 117*y^11 + 102*y^10 - 50*y^9 - 18*y^8 + 74*y^7 - 101*y^6 + 95*y^5 - 69*y^4 + 39*y^3 - 17*y^2 + 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 1)
\( x^{18} - 5 x^{17} + 14 x^{16} - 27 x^{15} + 44 x^{14} - 68 x^{13} + 98 x^{12} - 117 x^{11} + 102 x^{10} + \cdots - 1 \)
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: |
\(44055316944848498413\)
\(\medspace = 23^{7}\cdot 59^{3}\cdot 251^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.34\) | 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: | $23^{3/4}59^{1/2}251^{1/2}\approx 1278.0820739142598$ | ||
| Ramified primes: |
\(23\), \(59\), \(251\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1357}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{347}a^{17}-\frac{135}{347}a^{16}-\frac{133}{347}a^{15}-\frac{87}{347}a^{14}-\frac{97}{347}a^{13}+\frac{50}{347}a^{12}-\frac{156}{347}a^{11}+\frac{37}{347}a^{10}+\frac{150}{347}a^{9}-\frac{118}{347}a^{8}+\frac{54}{347}a^{7}-\frac{6}{347}a^{6}-\frac{15}{347}a^{5}-\frac{37}{347}a^{4}-\frac{117}{347}a^{3}-\frac{19}{347}a^{2}+\frac{24}{347}a+\frac{8}{347}$
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: |
$\frac{467}{347}a^{17}-\frac{1973}{347}a^{16}+\frac{4860}{347}a^{15}-\frac{8358}{347}a^{14}+\frac{12997}{347}a^{13}-\frac{20025}{347}a^{12}+\frac{27431}{347}a^{11}-\frac{28872}{347}a^{10}+\frac{19388}{347}a^{9}-\frac{2709}{347}a^{8}-\frac{14340}{347}a^{7}+\frac{25305}{347}a^{6}-\frac{26437}{347}a^{5}+\frac{19850}{347}a^{4}-\frac{11264}{347}a^{3}+\frac{4660}{347}a^{2}-\frac{937}{347}a-\frac{81}{347}$, $\frac{712}{347}a^{17}-\frac{3471}{347}a^{16}+\frac{8710}{347}a^{15}-\frac{15099}{347}a^{14}+\frac{22544}{347}a^{13}-\frac{34841}{347}a^{12}+\frac{48895}{347}a^{11}-\frac{51731}{347}a^{10}+\frac{32542}{347}a^{9}-\frac{389}{347}a^{8}-\frac{27829}{347}a^{7}+\frac{43961}{347}a^{6}-\frac{44339}{347}a^{5}+\frac{32646}{347}a^{4}-\frac{17374}{347}a^{3}+\frac{7639}{347}a^{2}-\frac{1650}{347}a+\frac{144}{347}$, $a$, $\frac{267}{347}a^{17}-\frac{651}{347}a^{16}+\frac{230}{347}a^{15}+\frac{2102}{347}a^{14}-\frac{5079}{347}a^{13}+\frac{7451}{347}a^{12}-\frac{12851}{347}a^{11}+\frac{24453}{347}a^{10}-\frac{34902}{347}a^{9}+\frac{30607}{347}a^{8}-\frac{11607}{347}a^{7}-\frac{9236}{347}a^{6}+\frac{23061}{347}a^{5}-\frac{27923}{347}a^{4}+\frac{23587}{347}a^{3}-\frac{13748}{347}a^{2}+\frac{5714}{347}a-\frac{1334}{347}$, $\frac{1074}{347}a^{17}-\frac{4802}{347}a^{16}+\frac{11920}{347}a^{15}-\frac{20221}{347}a^{14}+\frac{30458}{347}a^{13}-\frac{46583}{347}a^{12}+\frac{64946}{347}a^{11}-\frac{67485}{347}a^{10}+\frac{41038}{347}a^{9}+\frac{2005}{347}a^{8}-\frac{38817}{347}a^{7}+\frac{57404}{347}a^{6}-\frac{57056}{347}a^{5}+\frac{41460}{347}a^{4}-\frac{22946}{347}a^{3}+\frac{9436}{347}a^{2}-\frac{2678}{347}a+\frac{264}{347}$, $\frac{478}{347}a^{17}-\frac{2417}{347}a^{16}+\frac{6867}{347}a^{15}-\frac{13479}{347}a^{14}+\frac{21993}{347}a^{13}-\frac{33702}{347}a^{12}+\frac{48270}{347}a^{11}-\frac{58307}{347}a^{10}+\frac{51227}{347}a^{9}-\frac{24133}{347}a^{8}-\frac{12011}{347}a^{7}+\frac{40160}{347}a^{6}-\frac{51586}{347}a^{5}+\frac{47203}{347}a^{4}-\frac{32677}{347}a^{3}+\frac{16943}{347}a^{2}-\frac{6225}{347}a+\frac{1742}{347}$, $\frac{330}{347}a^{17}-\frac{1522}{347}a^{16}+\frac{3996}{347}a^{15}-\frac{7543}{347}a^{14}+\frac{12406}{347}a^{13}-\frac{19588}{347}a^{12}+\frac{27636}{347}a^{11}-\frac{32206}{347}a^{10}+\frac{27986}{347}a^{9}-\frac{14997}{347}a^{8}-\frac{3347}{347}a^{7}+\frac{20228}{347}a^{6}-\frac{28893}{347}a^{5}+\frac{27695}{347}a^{4}-\frac{20219}{347}a^{3}+\frac{11774}{347}a^{2}-\frac{4919}{347}a+\frac{1599}{347}$, $\frac{1132}{347}a^{17}-\frac{4998}{347}a^{16}+\frac{12881}{347}a^{15}-\frac{22491}{347}a^{14}+\frac{34895}{347}a^{13}-\frac{53052}{347}a^{12}+\frac{74636}{347}a^{11}-\frac{80954}{347}a^{10}+\frac{55984}{347}a^{9}-\frac{9003}{347}a^{8}-\frac{37420}{347}a^{7}+\frac{64690}{347}a^{6}-\frac{70765}{347}a^{5}+\frac{55276}{347}a^{4}-\frac{33202}{347}a^{3}+\frac{14233}{347}a^{2}-\frac{4756}{347}a+\frac{381}{347}$, $\frac{206}{347}a^{17}-\frac{1091}{347}a^{16}+\frac{2444}{347}a^{15}-\frac{3348}{347}a^{14}+\frac{3614}{347}a^{13}-\frac{5315}{347}a^{12}+\frac{7422}{347}a^{11}-\frac{3829}{347}a^{10}-\frac{7270}{347}a^{9}+\frac{17332}{347}a^{8}-\frac{16983}{347}a^{7}+\frac{9868}{347}a^{6}-\frac{1702}{347}a^{5}-\frac{5193}{347}a^{4}+\frac{8516}{347}a^{3}-\frac{7037}{347}a^{2}+\frac{3209}{347}a-\frac{1128}{347}$
| 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: | \( 226.057810023 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 226.057810023 \cdot 1}{2\cdot\sqrt{44055316944848498413}}\cr\approx \mathstrut & 0.165458580266 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 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 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 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 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 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 - 5*x^17 + 14*x^16 - 27*x^15 + 44*x^14 - 68*x^13 + 98*x^12 - 117*x^11 + 102*x^10 - 50*x^9 - 18*x^8 + 74*x^7 - 101*x^6 + 95*x^5 - 69*x^4 + 39*x^3 - 17*x^2 + 5*x - 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
$C_2\times S_4^3.S_4$ (as 18T912):
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 663552 |
| The 330 conjugacy class representatives for $C_2\times S_4^3.S_4$ |
| Character table for $C_2\times S_4^3.S_4$ |
Intermediate fields
| 3.1.23.1, 9.3.180181103.1 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | $18$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | R |
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 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.4.1 | $x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.4.2.1 | $x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(251\)
| $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |