Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490)
gp: K = bnfinit(y^15 - 6*y^14 + 26*y^13 - 74*y^12 + 169*y^11 - 242*y^10 + 94*y^9 + 786*y^8 - 2697*y^7 + 5514*y^6 - 7846*y^5 + 8590*y^4 - 6977*y^3 + 4350*y^2 - 1778*y + 490, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490)
\( x^{15} - 6 x^{14} + 26 x^{13} - 74 x^{12} + 169 x^{11} - 242 x^{10} + 94 x^{9} + 786 x^{8} - 2697 x^{7} + \cdots + 490 \)
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: |
\(-3428672784633475386368\)
\(\medspace = -\,2^{10}\cdot 443^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.27\) | 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: | $2^{2/3}443^{1/2}\approx 33.410927107861845$ | ||
| Ramified primes: |
\(2\), \(443\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-443}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{3}{8}a^{5}+\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{112}a^{13}+\frac{5}{112}a^{11}-\frac{1}{56}a^{10}+\frac{5}{56}a^{9}-\frac{1}{28}a^{7}+\frac{3}{56}a^{6}-\frac{29}{112}a^{5}-\frac{11}{56}a^{4}+\frac{23}{112}a^{3}-\frac{9}{28}a^{2}-\frac{23}{56}a+\frac{1}{8}$, $\frac{1}{74189920}a^{14}-\frac{3425}{2119712}a^{13}+\frac{1623501}{74189920}a^{12}-\frac{2644413}{74189920}a^{11}-\frac{59452}{2318435}a^{10}-\frac{575559}{5299280}a^{9}+\frac{1575307}{18547480}a^{8}+\frac{2838587}{37094960}a^{7}+\frac{466857}{74189920}a^{6}-\frac{1279419}{74189920}a^{5}+\frac{2950653}{14837984}a^{4}+\frac{2659171}{14837984}a^{3}+\frac{7085139}{37094960}a^{2}+\frac{68193}{1324820}a+\frac{11647}{151408}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $7$ | 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: |
$\frac{3499357}{74189920}a^{14}-\frac{66429}{302816}a^{13}+\frac{65459287}{74189920}a^{12}-\frac{155654641}{74189920}a^{11}+\frac{40070789}{9273740}a^{10}-\frac{20009973}{5299280}a^{9}-\frac{40507303}{9273740}a^{8}+\frac{1245185049}{37094960}a^{7}-\frac{5688599491}{74189920}a^{6}+\frac{9232576107}{74189920}a^{5}-\frac{2024702457}{14837984}a^{4}+\frac{1772107299}{14837984}a^{3}-\frac{2572048117}{37094960}a^{2}+\frac{80788297}{2649640}a-\frac{924913}{151408}$, $\frac{1945593}{74189920}a^{14}-\frac{227797}{2119712}a^{13}+\frac{31700223}{74189920}a^{12}-\frac{69256759}{74189920}a^{11}+\frac{34590557}{18547480}a^{10}-\frac{5891327}{5299280}a^{9}-\frac{14460661}{4636870}a^{8}+\frac{618285861}{37094960}a^{7}-\frac{2501363619}{74189920}a^{6}+\frac{3734161873}{74189920}a^{5}-\frac{730456901}{14837984}a^{4}+\frac{555968297}{14837984}a^{3}-\frac{605744193}{37094960}a^{2}+\frac{1767786}{331205}a+\frac{89389}{151408}$, $\frac{881633}{74189920}a^{14}-\frac{101133}{2119712}a^{13}+\frac{13751793}{74189920}a^{12}-\frac{27249489}{74189920}a^{11}+\frac{6301531}{9273740}a^{10}-\frac{197747}{5299280}a^{9}-\frac{39034769}{18547480}a^{8}+\frac{307726711}{37094960}a^{7}-\frac{1000104039}{74189920}a^{6}+\frac{1175348513}{74189920}a^{5}-\frac{83642479}{14837984}a^{4}-\frac{50302369}{14837984}a^{3}+\frac{494655167}{37094960}a^{2}-\frac{1849773}{189260}a+\frac{1062403}{151408}$, $\frac{1384679}{74189920}a^{14}-\frac{162051}{2119712}a^{13}+\frac{22159609}{74189920}a^{12}-\frac{45480817}{74189920}a^{11}+\frac{2710097}{2318435}a^{10}-\frac{1483011}{5299280}a^{9}-\frac{56053867}{18547480}a^{8}+\frac{485074253}{37094960}a^{7}-\frac{1668108297}{74189920}a^{6}+\frac{2174482259}{74189920}a^{5}-\frac{240729519}{14837984}a^{4}+\frac{39130827}{14837984}a^{3}+\frac{653144341}{37094960}a^{2}-\frac{9058269}{662410}a+\frac{1666199}{151408}$, $\frac{142363}{7418992}a^{14}+\frac{5241}{1059856}a^{13}-\frac{167035}{7418992}a^{12}+\frac{4086949}{7418992}a^{11}-\frac{2213663}{1854748}a^{10}+\frac{2000343}{529928}a^{9}-\frac{1631134}{463687}a^{8}-\frac{4190461}{3709496}a^{7}+\frac{187187843}{7418992}a^{6}-\frac{380597557}{7418992}a^{5}+\frac{539114577}{7418992}a^{4}-\frac{455037643}{7418992}a^{3}+\frac{142285975}{3709496}a^{2}-\frac{2863053}{264964}a+\frac{50709}{75704}$, $\frac{890807}{14837984}a^{14}-\frac{638171}{2119712}a^{13}+\frac{17833023}{14837984}a^{12}-\frac{43418279}{14837984}a^{11}+\frac{21936529}{3709496}a^{10}-\frac{795333}{151408}a^{9}-\frac{12495235}{1854748}a^{8}+\frac{357677715}{7418992}a^{7}-\frac{1629325517}{14837984}a^{6}+\frac{2544667831}{14837984}a^{5}-\frac{2561313921}{14837984}a^{4}+\frac{1896547673}{14837984}a^{3}-\frac{341884727}{7418992}a^{2}+\frac{1457927}{264964}a+\frac{1947949}{151408}$, $\frac{107505}{14837984}a^{14}-\frac{166809}{2119712}a^{13}+\frac{4542953}{14837984}a^{12}-\frac{14368065}{14837984}a^{11}+\frac{935481}{463687}a^{10}-\frac{3234579}{1059856}a^{9}+\frac{976025}{3709496}a^{8}+\frac{85660439}{7418992}a^{7}-\frac{546374863}{14837984}a^{6}+\frac{980564425}{14837984}a^{5}-\frac{1231661195}{14837984}a^{4}+\frac{1107621355}{14837984}a^{3}-\frac{352570265}{7418992}a^{2}+\frac{5288089}{264964}a-\frac{659201}{151408}$
| 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: | \( 191156.293017 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 191156.293017 \cdot 1}{2\cdot\sqrt{3428672784633475386368}}\cr\approx \mathstrut & 1.26207353036 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490)
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 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490, 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 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490);
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 - 6*x^14 + 26*x^13 - 74*x^12 + 169*x^11 - 242*x^10 + 94*x^9 + 786*x^8 - 2697*x^7 + 5514*x^6 - 7846*x^5 + 8590*x^4 - 6977*x^3 + 4350*x^2 - 1778*x + 490);
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
| 3.1.1772.1, 5.1.196249.1 |
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 | $15$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(443\)
| $\Q_{443}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |