Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(y^18 - 5*y^17 + 12*y^16 - 22*y^15 + 39*y^14 - 63*y^13 + 85*y^12 - 97*y^11 + 99*y^10 - 98*y^9 + 91*y^8 - 71*y^7 + 53*y^6 - 44*y^5 + 29*y^4 - 15*y^3 + 10*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*x^2 - 4*x + 1)
\( x^{18} - 5 x^{17} + 12 x^{16} - 22 x^{15} + 39 x^{14} - 63 x^{13} + 85 x^{12} - 97 x^{11} + 99 x^{10} + \cdots + 1 \)
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: |
\(7182890828838813137\)
\(\medspace = 7^{12}\cdot 113\cdot 2143^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.16\) | 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: | $7^{2/3}113^{1/2}2143^{1/2}\approx 1800.7316613040525$ | ||
| Ramified primes: |
\(7\), \(113\), \(2143\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{113}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{639241}a^{17}+\frac{230257}{639241}a^{16}+\frac{149565}{639241}a^{15}+\frac{27133}{639241}a^{14}-\frac{242649}{639241}a^{13}+\frac{15504}{639241}a^{12}-\frac{178852}{639241}a^{11}+\frac{282104}{639241}a^{10}+\frac{78650}{639241}a^{9}-\frac{230569}{639241}a^{8}+\frac{243027}{639241}a^{7}+\frac{86622}{639241}a^{6}+\frac{157335}{639241}a^{5}-\frac{72708}{639241}a^{4}-\frac{167677}{639241}a^{3}-\frac{124230}{639241}a^{2}-\frac{52741}{639241}a+\frac{52372}{639241}$
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: |
$\frac{80431}{639241}a^{17}-\frac{289485}{639241}a^{16}+\frac{425377}{639241}a^{15}-\frac{673692}{639241}a^{14}+\frac{1443734}{639241}a^{13}-\frac{2074690}{639241}a^{12}+\frac{2151975}{639241}a^{11}-\frac{2509435}{639241}a^{10}+\frac{3165419}{639241}a^{9}-\frac{3710034}{639241}a^{8}+\frac{3389544}{639241}a^{7}-\frac{2550541}{639241}a^{6}+\frac{3392754}{639241}a^{5}-\frac{2757444}{639241}a^{4}+\frac{277831}{639241}a^{3}-\frac{606300}{639241}a^{2}+\frac{1270387}{639241}a+\frac{373383}{639241}$, $\frac{418297}{639241}a^{17}-\frac{1825346}{639241}a^{16}+\frac{3909581}{639241}a^{15}-\frac{7103105}{639241}a^{14}+\frac{12761288}{639241}a^{13}-\frac{19639728}{639241}a^{12}+\frac{25245790}{639241}a^{11}-\frac{28119075}{639241}a^{10}+\frac{28008348}{639241}a^{9}-\frac{27043999}{639241}a^{8}+\frac{23259947}{639241}a^{7}-\frac{16995135}{639241}a^{6}+\frac{14403883}{639241}a^{5}-\frac{11236316}{639241}a^{4}+\frac{5128861}{639241}a^{3}-\frac{3692384}{639241}a^{2}+\frac{2640279}{639241}a-\frac{377827}{639241}$, $a$, $\frac{152543}{639241}a^{17}-\frac{920917}{639241}a^{16}+\frac{2500228}{639241}a^{15}-\frac{4610943}{639241}a^{14}+\frac{7875349}{639241}a^{13}-\frac{12949848}{639241}a^{12}+\frac{18083992}{639241}a^{11}-\frac{20530119}{639241}a^{10}+\frac{20048333}{639241}a^{9}-\frac{19185136}{639241}a^{8}+\frac{18463096}{639241}a^{7}-\frac{14873508}{639241}a^{6}+\frac{9638175}{639241}a^{5}-\frac{7296745}{639241}a^{4}+\frac{5750691}{639241}a^{3}-\frac{2674409}{639241}a^{2}+\frac{216863}{639241}a-\frac{252022}{639241}$, $\frac{43345}{639241}a^{17}+\frac{19932}{639241}a^{16}-\frac{287297}{639241}a^{15}+\frac{515686}{639241}a^{14}-\frac{827973}{639241}a^{13}+\frac{1457071}{639241}a^{12}-\frac{2182056}{639241}a^{11}+\frac{2313755}{639241}a^{10}-\frac{1266485}{639241}a^{9}-\frac{119511}{639241}a^{8}+\frac{592117}{639241}a^{7}-\frac{1549526}{639241}a^{6}+\frac{2180310}{639241}a^{5}-\frac{1987853}{639241}a^{4}+\frac{1489087}{639241}a^{3}-\frac{1061648}{639241}a^{2}+\frac{1145653}{639241}a-\frac{519692}{639241}$, $\frac{141184}{639241}a^{17}-\frac{636008}{639241}a^{16}+\frac{1415489}{639241}a^{15}-\frac{2782805}{639241}a^{14}+\frac{5161184}{639241}a^{13}-\frac{8154581}{639241}a^{12}+\frac{11124311}{639241}a^{11}-\frac{13402671}{639241}a^{10}+\frac{14568732}{639241}a^{9}-\frac{14647555}{639241}a^{8}+\frac{13048113}{639241}a^{7}-\frac{10546220}{639241}a^{6}+\frac{9148505}{639241}a^{5}-\frac{6666704}{639241}a^{4}+\frac{4177072}{639241}a^{3}-\frac{2989967}{639241}a^{2}+\frac{2250788}{639241}a-\frac{651440}{639241}$, $\frac{374956}{639241}a^{17}-\frac{1563491}{639241}a^{16}+\frac{2877415}{639241}a^{15}-\frac{4314054}{639241}a^{14}+\frac{7504737}{639241}a^{13}-\frac{11446168}{639241}a^{12}+\frac{12649136}{639241}a^{11}-\frac{10766425}{639241}a^{10}+\frac{9133721}{639241}a^{9}-\frac{10587257}{639241}a^{8}+\frac{10215877}{639241}a^{7}-\frac{4871265}{639241}a^{6}+\frac{3264298}{639241}a^{5}-\frac{5064608}{639241}a^{4}+\frac{2329825}{639241}a^{3}+\frac{68549}{639241}a^{2}+\frac{1283662}{639241}a-\frac{287888}{639241}$, $\frac{248192}{639241}a^{17}-\frac{839297}{639241}a^{16}+\frac{1390092}{639241}a^{15}-\frac{2767363}{639241}a^{14}+\frac{5746412}{639241}a^{13}-\frac{8572185}{639241}a^{12}+\frac{10765794}{639241}a^{11}-\frac{13534823}{639241}a^{10}+\frac{16418649}{639241}a^{9}-\frac{17786435}{639241}a^{8}+\frac{14557449}{639241}a^{7}-\frac{10932985}{639241}a^{6}+\frac{12118932}{639241}a^{5}-\frac{9359121}{639241}a^{4}+\frac{3613044}{639241}a^{3}-\frac{3577212}{639241}a^{2}+\frac{3039890}{639241}a-\frac{654311}{639241}$, $\frac{256051}{639241}a^{17}-\frac{940805}{639241}a^{16}+\frac{1896469}{639241}a^{15}-\frac{3674851}{639241}a^{14}+\frac{6902306}{639241}a^{13}-\frac{10739003}{639241}a^{12}+\frac{14055090}{639241}a^{11}-\frac{16563480}{639241}a^{10}+\frac{18300675}{639241}a^{9}-\frac{18858453}{639241}a^{8}+\frac{15733016}{639241}a^{7}-\frac{11601593}{639241}a^{6}+\frac{9765639}{639241}a^{5}-\frac{8011357}{639241}a^{4}+\frac{4593684}{639241}a^{3}-\frac{2501293}{639241}a^{2}+\frac{2137298}{639241}a-\frac{733967}{639241}$
| 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: | \( 84.7255286435 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 84.7255286435 \cdot 1}{2\cdot\sqrt{7182890828838813137}}\cr\approx \mathstrut & 0.153579622923 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*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 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*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 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*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 - 5*x^17 + 12*x^16 - 22*x^15 + 39*x^14 - 63*x^13 + 85*x^12 - 97*x^11 + 99*x^10 - 98*x^9 + 91*x^8 - 71*x^7 + 53*x^6 - 44*x^5 + 29*x^4 - 15*x^3 + 10*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
$C_2\times S_4^3.A_4$ (as 18T879):
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 331776 |
| The 360 conjugacy class representatives for $C_2\times S_4^3.A_4$ |
| Character table for $C_2\times S_4^3.A_4$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.3.252121807.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}$ | $18$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | $18$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |
|
\(113\)
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
|
\(2143\)
| $\Q_{2143}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2143}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |