Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972)
gp: K = bnfinit(y^15 - 3*y^14 + 18*y^13 - 46*y^12 + 134*y^11 - 262*y^10 + 548*y^9 - 748*y^8 + 1117*y^7 - 883*y^6 + 422*y^5 + 1266*y^4 - 2232*y^3 + 4104*y^2 - 3240*y + 972, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972)
\( x^{15} - 3 x^{14} + 18 x^{13} - 46 x^{12} + 134 x^{11} - 262 x^{10} + 548 x^{9} - 748 x^{8} + 1117 x^{7} + \cdots + 972 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-77027586128169029561344\)
\(\medspace = -\,2^{10}\cdot 691^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.56\) | 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: | $2^{2/3}691^{1/2}\approx 41.72781914927636$ | ||
| Ramified primes: |
\(2\), \(691\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-691}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{3}a$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{36}a^{12}-\frac{1}{36}a^{9}-\frac{1}{36}a^{8}+\frac{2}{9}a^{7}+\frac{2}{9}a^{6}+\frac{17}{36}a^{5}+\frac{5}{18}a^{4}-\frac{1}{9}a^{3}-\frac{5}{18}a^{2}-\frac{1}{3}a$, $\frac{1}{7128}a^{13}-\frac{1}{792}a^{12}+\frac{7}{792}a^{11}+\frac{251}{7128}a^{10}+\frac{437}{7128}a^{9}+\frac{203}{7128}a^{8}-\frac{1039}{7128}a^{7}-\frac{1687}{7128}a^{6}+\frac{1385}{3564}a^{5}+\frac{1513}{3564}a^{4}+\frac{13}{891}a^{3}+\frac{259}{594}a^{2}-\frac{13}{33}a+\frac{1}{66}$, $\frac{1}{35933523912}a^{14}+\frac{13351}{230343102}a^{13}-\frac{5336395}{1996306884}a^{12}+\frac{515561}{74242818}a^{11}+\frac{741067471}{17966761956}a^{10}+\frac{328138090}{4491690489}a^{9}-\frac{356625056}{4491690489}a^{8}+\frac{110551759}{4491690489}a^{7}-\frac{72175715}{2764117224}a^{6}-\frac{1162404212}{4491690489}a^{5}+\frac{1453581595}{17966761956}a^{4}-\frac{395266601}{998153442}a^{3}+\frac{169231111}{998153442}a^{2}+\frac{266489}{30247074}a-\frac{5297423}{110905938}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}$, which has order $2$
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: | $7$ | 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: |
$\frac{16906271}{17966761956}a^{14}-\frac{4513811}{921372408}a^{13}+\frac{75318835}{3992613768}a^{12}-\frac{201758207}{3266683992}a^{11}+\frac{5142102635}{35933523912}a^{10}-\frac{9999998215}{35933523912}a^{9}+\frac{18015710411}{35933523912}a^{8}-\frac{24957369763}{35933523912}a^{7}+\frac{1875849787}{2764117224}a^{6}-\frac{3531183049}{8983380978}a^{5}-\frac{2358900161}{17966761956}a^{4}+\frac{799794700}{499076721}a^{3}-\frac{991628224}{499076721}a^{2}+\frac{10207480}{15123537}a-\frac{58921847}{110905938}$, $\frac{706529}{921372408}a^{14}-\frac{518395}{307124136}a^{13}+\frac{1912115}{102374712}a^{12}-\frac{3645091}{83761128}a^{11}+\frac{158953441}{921372408}a^{10}-\frac{317050565}{921372408}a^{9}+\frac{734326033}{921372408}a^{8}-\frac{996274931}{921372408}a^{7}+\frac{893435737}{460686204}a^{6}-\frac{655104145}{460686204}a^{5}+\frac{130073855}{115171551}a^{4}+\frac{24022583}{12796839}a^{3}-\frac{47003690}{12796839}a^{2}+\frac{6556585}{775566}a-\frac{5083634}{1421871}$, $\frac{13157959}{17966761956}a^{14}-\frac{1882267}{921372408}a^{13}+\frac{25705391}{3992613768}a^{12}-\frac{38421481}{3266683992}a^{11}-\frac{111245735}{35933523912}a^{10}+\frac{2185129213}{35933523912}a^{9}-\frac{6169509029}{35933523912}a^{8}+\frac{15972220699}{35933523912}a^{7}-\frac{2446014475}{2764117224}a^{6}+\frac{10501883065}{8983380978}a^{5}-\frac{23539139803}{17966761956}a^{4}+\frac{776390189}{998153442}a^{3}+\frac{705515563}{499076721}a^{2}-\frac{4776365}{1374867}a+\frac{183191549}{110905938}$, $\frac{7624957}{3992613768}a^{14}-\frac{1932125}{307124136}a^{13}+\frac{16937015}{443623752}a^{12}-\frac{35719913}{362964888}a^{11}+\frac{1140053737}{3992613768}a^{10}-\frac{2235084155}{3992613768}a^{9}+\frac{4150357993}{3992613768}a^{8}-\frac{5836049981}{3992613768}a^{7}+\frac{29866475}{17062452}a^{6}-\frac{172353481}{166358907}a^{5}-\frac{566916208}{499076721}a^{4}+\frac{1800833965}{499076721}a^{3}-\frac{2432719073}{332717814}a^{2}+\frac{23321965}{3360786}a-\frac{38319739}{18484323}$, $\frac{7651249}{5988920652}a^{14}-\frac{126811}{38390517}a^{13}+\frac{8104637}{332717814}a^{12}-\frac{2649631}{49495212}a^{11}+\frac{527491999}{2994460326}a^{10}-\frac{1885583473}{5988920652}a^{9}+\frac{1896774247}{2994460326}a^{8}-\frac{4927808905}{5988920652}a^{7}+\frac{531050161}{460686204}a^{6}-\frac{3688338373}{5988920652}a^{5}+\frac{755928001}{2994460326}a^{4}+\frac{1322985607}{998153442}a^{3}-\frac{749569349}{332717814}a^{2}+\frac{11714849}{5041179}a-\frac{21163735}{18484323}$, $\frac{373237}{230343102}a^{14}-\frac{683651}{307124136}a^{13}+\frac{2455477}{102374712}a^{12}-\frac{273157}{7614648}a^{11}+\frac{128895779}{921372408}a^{10}-\frac{179370457}{921372408}a^{9}+\frac{449829605}{921372408}a^{8}-\frac{363458671}{921372408}a^{7}+\frac{845298733}{921372408}a^{6}-\frac{25240241}{460686204}a^{5}-\frac{53104643}{460686204}a^{4}+\frac{14474008}{12796839}a^{3}-\frac{68578667}{25593678}a^{2}+\frac{764575}{387783}a-\frac{4962545}{2843742}$, $\frac{8586443}{17966761956}a^{14}-\frac{102577}{115171551}a^{13}+\frac{6400943}{1996306884}a^{12}-\frac{17801335}{1633341996}a^{11}-\frac{17466719}{17966761956}a^{10}-\frac{650262425}{17966761956}a^{9}-\frac{101557409}{17966761956}a^{8}-\frac{1113561689}{17966761956}a^{7}-\frac{123461531}{691029306}a^{6}-\frac{7938234179}{17966761956}a^{5}-\frac{5613441109}{8983380978}a^{4}-\frac{216081803}{332717814}a^{3}+\frac{220710559}{998153442}a^{2}+\frac{10168018}{15123537}a-\frac{23770651}{55452969}$
| 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: | \( 817777.771658 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 817777.771658 \cdot 2}{2\cdot\sqrt{77027586128169029561344}}\cr\approx \mathstrut & 2.27824922751 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972)
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^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972, 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^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972);
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^15 - 3*x^14 + 18*x^13 - 46*x^12 + 134*x^11 - 262*x^10 + 548*x^9 - 748*x^8 + 1117*x^7 - 883*x^6 + 422*x^5 + 1266*x^4 - 2232*x^3 + 4104*x^2 - 3240*x + 972);
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 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.2764.1, 5.1.477481.1 |
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
| Galois closure: | deg 30 |
| 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 | R | ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(691\)
| $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |