Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(y^18 - 4*y^17 + 11*y^16 - 22*y^15 + 36*y^14 - 53*y^13 + 80*y^12 - 116*y^11 + 154*y^10 - 173*y^9 + 154*y^8 - 116*y^7 + 80*y^6 - 53*y^5 + 36*y^4 - 22*y^3 + 11*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*x + 1)
\( x^{18} - 4 x^{17} + 11 x^{16} - 22 x^{15} + 36 x^{14} - 53 x^{13} + 80 x^{12} - 116 x^{11} + 154 x^{10} + \cdots + 1 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10383286663954563863\)
\(\medspace = -\,7^{8}\cdot 23^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.39\) | 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: | $7^{2/3}23^{1/2}\approx 17.5494136775664$ | ||
| Ramified primes: |
\(7\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1597}a^{16}+\frac{210}{1597}a^{15}+\frac{234}{1597}a^{14}+\frac{337}{1597}a^{13}+\frac{55}{1597}a^{12}+\frac{201}{1597}a^{11}-\frac{80}{1597}a^{10}+\frac{130}{1597}a^{9}-\frac{692}{1597}a^{8}+\frac{130}{1597}a^{7}-\frac{80}{1597}a^{6}+\frac{201}{1597}a^{5}+\frac{55}{1597}a^{4}+\frac{337}{1597}a^{3}+\frac{234}{1597}a^{2}+\frac{210}{1597}a+\frac{1}{1597}$, $\frac{1}{11179}a^{17}+\frac{2}{11179}a^{16}-\frac{327}{11179}a^{15}+\frac{2769}{11179}a^{14}+\frac{1824}{11179}a^{13}+\frac{4731}{11179}a^{12}-\frac{1963}{11179}a^{11}-\frac{3991}{11179}a^{10}+\frac{4208}{11179}a^{9}+\frac{1933}{11179}a^{8}-\frac{3165}{11179}a^{7}+\frac{2468}{11179}a^{6}-\frac{1828}{11179}a^{5}-\frac{3118}{11179}a^{4}-\frac{4385}{11179}a^{3}+\frac{1045}{11179}a^{2}+\frac{2634}{11179}a-\frac{1805}{11179}$
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: | $8$ | 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: |
$\frac{5140}{11179}a^{17}-\frac{22942}{11179}a^{16}+\frac{62220}{11179}a^{15}-\frac{125719}{11179}a^{14}+\frac{202945}{11179}a^{13}-\frac{292672}{11179}a^{12}+\frac{437039}{11179}a^{11}-\frac{640320}{11179}a^{10}+\frac{843593}{11179}a^{9}-\frac{935947}{11179}a^{8}+\frac{798467}{11179}a^{7}-\frac{530900}{11179}a^{6}+\frac{348409}{11179}a^{5}-\frac{213268}{11179}a^{4}+\frac{137656}{11179}a^{3}-\frac{99774}{11179}a^{2}+\frac{33604}{11179}a-\frac{9994}{11179}$, $\frac{5956}{11179}a^{17}-\frac{19203}{11179}a^{16}+\frac{47815}{11179}a^{15}-\frac{78495}{11179}a^{14}+\frac{109693}{11179}a^{13}-\frac{139509}{11179}a^{12}+\frac{220132}{11179}a^{11}-\frac{309392}{11179}a^{10}+\frac{370285}{11179}a^{9}-\frac{280071}{11179}a^{8}+\frac{77155}{11179}a^{7}+\frac{62380}{11179}a^{6}-\frac{60171}{11179}a^{5}-\frac{3427}{11179}a^{4}-\frac{13948}{11179}a^{3}+\frac{38652}{11179}a^{2}-\frac{24005}{11179}a+\frac{17219}{11179}$, $\frac{1478}{11179}a^{17}-\frac{11380}{11179}a^{16}+\frac{27519}{11179}a^{15}-\frac{55751}{11179}a^{14}+\frac{78082}{11179}a^{13}-\frac{101018}{11179}a^{12}+\frac{142020}{11179}a^{11}-\frac{235502}{11179}a^{10}+\frac{286588}{11179}a^{9}-\frac{279586}{11179}a^{8}+\frac{132313}{11179}a^{7}+\frac{32330}{11179}a^{6}-\frac{16178}{11179}a^{5}-\frac{30964}{11179}a^{4}+\frac{23244}{11179}a^{3}+\frac{884}{11179}a^{2}-\frac{23007}{11179}a+\frac{12013}{11179}$, $\frac{2878}{11179}a^{17}-\frac{11436}{11179}a^{16}+\frac{43143}{11179}a^{15}-\frac{100544}{11179}a^{14}+\frac{193582}{11179}a^{13}-\frac{308561}{11179}a^{12}+\frac{452939}{11179}a^{11}-\frac{642125}{11179}a^{10}+\frac{932464}{11179}a^{9}-\frac{1186534}{11179}a^{8}+\frac{1266082}{11179}a^{7}-\frac{1090960}{11179}a^{6}+\frac{662605}{11179}a^{5}-\frac{405835}{11179}a^{4}+\frac{311091}{11179}a^{3}-\frac{199351}{11179}a^{2}+\frac{113577}{11179}a-\frac{36105}{11179}$, $\frac{27714}{11179}a^{17}-\frac{96423}{11179}a^{16}+\frac{243427}{11179}a^{15}-\frac{446022}{11179}a^{14}+\frac{673313}{11179}a^{13}-\frac{954864}{11179}a^{12}+\frac{1477858}{11179}a^{11}-\frac{2106693}{11179}a^{10}+\frac{2640912}{11179}a^{9}-\frac{2672095}{11179}a^{8}+\frac{1953433}{11179}a^{7}-\frac{1306406}{11179}a^{6}+\frac{937691}{11179}a^{5}-\frac{603340}{11179}a^{4}+\frac{407138}{11179}a^{3}-\frac{211153}{11179}a^{2}+\frac{71857}{11179}a-\frac{26627}{11179}$, $\frac{6873}{11179}a^{17}-\frac{26588}{11179}a^{16}+\frac{81303}{11179}a^{15}-\frac{166086}{11179}a^{14}+\frac{285274}{11179}a^{13}-\frac{422099}{11179}a^{12}+\frac{625019}{11179}a^{11}-\frac{895208}{11179}a^{10}+\frac{1241911}{11179}a^{9}-\frac{1428890}{11179}a^{8}+\frac{1331121}{11179}a^{7}-\frac{994921}{11179}a^{6}+\frac{580301}{11179}a^{5}-\frac{407243}{11179}a^{4}+\frac{325776}{11179}a^{3}-\frac{187756}{11179}a^{2}+\frac{97655}{11179}a-\frac{26230}{11179}$, $\frac{6815}{11179}a^{17}-\frac{15945}{11179}a^{16}+\frac{34427}{11179}a^{15}-\frac{44882}{11179}a^{14}+\frac{49121}{11179}a^{13}-\frac{60036}{11179}a^{12}+\frac{117861}{11179}a^{11}-\frac{138254}{11179}a^{10}+\frac{116001}{11179}a^{9}+\frac{12863}{11179}a^{8}-\frac{183222}{11179}a^{7}+\frac{136404}{11179}a^{6}-\frac{57656}{11179}a^{5}+\frac{29976}{11179}a^{4}-\frac{19773}{11179}a^{3}+\frac{55798}{11179}a^{2}-\frac{31527}{11179}a+\frac{10966}{11179}$, $\frac{8937}{11179}a^{17}-\frac{34570}{11179}a^{16}+\frac{82827}{11179}a^{15}-\frac{146434}{11179}a^{14}+\frac{203699}{11179}a^{13}-\frac{266565}{11179}a^{12}+\frac{410696}{11179}a^{11}-\frac{606828}{11179}a^{10}+\frac{717666}{11179}a^{9}-\frac{618144}{11179}a^{8}+\frac{255852}{11179}a^{7}+\frac{26102}{11179}a^{6}+\frac{7415}{11179}a^{5}-\frac{52452}{11179}a^{4}+\frac{16379}{11179}a^{3}+\frac{29704}{11179}a^{2}-\frac{38378}{11179}a+\frac{25821}{11179}$
| 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: | \( 77.8161636269 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 77.8161636269 \cdot 1}{2\cdot\sqrt{10383286663954563863}}\cr\approx \mathstrut & 0.184285520456 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*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 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*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 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*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 - 4*x^17 + 11*x^16 - 22*x^15 + 36*x^14 - 53*x^13 + 80*x^12 - 116*x^11 + 154*x^10 - 173*x^9 + 154*x^8 - 116*x^7 + 80*x^6 - 53*x^5 + 36*x^4 - 22*x^3 + 11*x^2 - 4*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
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 54 |
| The 10 conjugacy class representatives for $C_9:C_6$ |
| Character table for $C_9:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.671898241.1 x3 |
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 9 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| Minimal sibling: | 9.1.671898241.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |