Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43)
gp: K = bnfinit(y^15 - 4*y^13 - 11*y^12 + 9*y^11 + 30*y^10 + 5*y^9 + 7*y^8 - 32*y^7 - 7*y^6 - 78*y^5 - 120*y^4 + 36*y^3 + 154*y^2 + 134*y + 43, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43)
\( x^{15} - 4 x^{13} - 11 x^{12} + 9 x^{11} + 30 x^{10} + 5 x^{9} + 7 x^{8} - 32 x^{7} - 7 x^{6} + \cdots + 43 \)
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: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-982108001708984375\)
\(\medspace = -\,5^{16}\cdot 23^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.83\) | 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: | $5^{48/25}23^{1/2}\approx 105.41109538118516$ | ||
| Ramified primes: |
\(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{24412247885}a^{14}+\frac{1646829922}{24412247885}a^{13}-\frac{1315688103}{24412247885}a^{12}-\frac{356133432}{4882449577}a^{11}-\frac{435577617}{24412247885}a^{10}-\frac{369787707}{4882449577}a^{9}+\frac{6165042833}{24412247885}a^{8}-\frac{8742388874}{24412247885}a^{7}-\frac{5629932023}{24412247885}a^{6}-\frac{10156258087}{24412247885}a^{5}-\frac{1463424901}{24412247885}a^{4}+\frac{2623546591}{24412247885}a^{3}-\frac{4022754281}{24412247885}a^{2}-\frac{7981696253}{24412247885}a-\frac{23277162}{113545339}$
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: |
$\frac{20062813}{567726695}a^{14}-\frac{1609591}{567726695}a^{13}-\frac{63103153}{567726695}a^{12}-\frac{188726017}{567726695}a^{11}+\frac{32804232}{113545339}a^{10}+\frac{367824721}{567726695}a^{9}-\frac{104556667}{567726695}a^{8}+\frac{388923204}{567726695}a^{7}-\frac{420738634}{567726695}a^{6}+\frac{385633859}{567726695}a^{5}-\frac{1034458567}{567726695}a^{4}-\frac{1719977241}{567726695}a^{3}+\frac{1193174627}{567726695}a^{2}+\frac{1091903873}{567726695}a+\frac{769554099}{567726695}$, $\frac{413508427}{24412247885}a^{14}-\frac{545172210}{4882449577}a^{13}-\frac{3475553109}{24412247885}a^{12}+\frac{338315801}{4882449577}a^{11}+\frac{33918122949}{24412247885}a^{10}+\frac{14740474426}{24412247885}a^{9}-\frac{35319244184}{24412247885}a^{8}-\frac{15177723731}{24412247885}a^{7}-\frac{91968070336}{24412247885}a^{6}-\frac{3880417043}{4882449577}a^{5}-\frac{121415319761}{24412247885}a^{4}+\frac{40249491276}{24412247885}a^{3}+\frac{256967725433}{24412247885}a^{2}+\frac{218302134424}{24412247885}a+\frac{2094054602}{567726695}$, $\frac{2208170669}{24412247885}a^{14}-\frac{2718425511}{24412247885}a^{13}-\frac{8903130271}{24412247885}a^{12}-\frac{3266060709}{4882449577}a^{11}+\frac{46953893333}{24412247885}a^{10}+\frac{48088917457}{24412247885}a^{9}-\frac{35650998203}{24412247885}a^{8}+\frac{17510871003}{24412247885}a^{7}-\frac{126224330636}{24412247885}a^{6}+\frac{28221603571}{24412247885}a^{5}-\frac{256765705062}{24412247885}a^{4}-\frac{95664317312}{24412247885}a^{3}+\frac{294645582699}{24412247885}a^{2}+\frac{342941515869}{24412247885}a+\frac{4319828461}{567726695}$, $\frac{1651389291}{24412247885}a^{14}-\frac{397735793}{24412247885}a^{13}-\frac{4837162954}{24412247885}a^{12}-\frac{15924278448}{24412247885}a^{11}+\frac{14492059128}{24412247885}a^{10}+\frac{26153058728}{24412247885}a^{9}-\frac{318737857}{24412247885}a^{8}+\frac{37888778649}{24412247885}a^{7}-\frac{40289554576}{24412247885}a^{6}+\frac{52243836821}{24412247885}a^{5}-\frac{131209955638}{24412247885}a^{4}-\frac{107838708889}{24412247885}a^{3}+\frac{19815017063}{24412247885}a^{2}+\frac{73694264452}{24412247885}a+\frac{1247851591}{567726695}$, $\frac{1334201126}{24412247885}a^{14}-\frac{131591827}{24412247885}a^{13}-\frac{3254866046}{24412247885}a^{12}-\frac{13953612931}{24412247885}a^{11}+\frac{7482413911}{24412247885}a^{10}+\frac{3159122374}{4882449577}a^{9}+\frac{10651952259}{24412247885}a^{8}+\frac{40978425514}{24412247885}a^{7}-\frac{26583751922}{24412247885}a^{6}+\frac{63541427757}{24412247885}a^{5}-\frac{115544080461}{24412247885}a^{4}-\frac{68706540664}{24412247885}a^{3}-\frac{14257945693}{4882449577}a^{2}+\frac{11548116264}{24412247885}a+\frac{820821207}{567726695}$, $\frac{982258969}{24412247885}a^{14}+\frac{825088044}{24412247885}a^{13}-\frac{2474356559}{24412247885}a^{12}-\frac{12260598544}{24412247885}a^{11}-\frac{2718820204}{24412247885}a^{10}+\frac{18137305231}{24412247885}a^{9}+\frac{3482641449}{4882449577}a^{8}+\frac{28917031994}{24412247885}a^{7}+\frac{4960425342}{24412247885}a^{6}+\frac{22449556132}{24412247885}a^{5}-\frac{50924267443}{24412247885}a^{4}-\frac{119040091844}{24412247885}a^{3}-\frac{104652572552}{24412247885}a^{2}+\frac{4479001331}{24412247885}a+\frac{419930387}{567726695}$, $\frac{3744897011}{24412247885}a^{14}-\frac{116117682}{4882449577}a^{13}-\frac{11735355329}{24412247885}a^{12}-\frac{7226803207}{4882449577}a^{11}+\frac{33530260923}{24412247885}a^{10}+\frac{14028447460}{4882449577}a^{9}-\frac{8033837362}{24412247885}a^{8}+\frac{64863529691}{24412247885}a^{7}-\frac{92901831181}{24412247885}a^{6}+\frac{18027217301}{4882449577}a^{5}-\frac{264082013466}{24412247885}a^{4}-\frac{284185081042}{24412247885}a^{3}+\frac{18795357474}{4882449577}a^{2}+\frac{243960829309}{24412247885}a+\frac{4016251854}{567726695}$, $\frac{2400247711}{24412247885}a^{14}+\frac{1475535691}{24412247885}a^{13}-\frac{7034658861}{24412247885}a^{12}-\frac{27325732651}{24412247885}a^{11}+\frac{2528403861}{24412247885}a^{10}+\frac{49170664164}{24412247885}a^{9}+\frac{14803993657}{24412247885}a^{8}+\frac{49013750274}{24412247885}a^{7}+\frac{3941636956}{24412247885}a^{6}+\frac{49186451976}{24412247885}a^{5}-\frac{95164769357}{24412247885}a^{4}-\frac{272339936194}{24412247885}a^{3}-\frac{14528769293}{4882449577}a^{2}+\frac{94275222727}{24412247885}a+\frac{172067607}{113545339}$, $\frac{825088044}{24412247885}a^{14}+\frac{1454679317}{24412247885}a^{13}-\frac{291149977}{4882449577}a^{12}-\frac{2311830185}{4882449577}a^{11}-\frac{11330463839}{24412247885}a^{10}+\frac{2500382480}{4882449577}a^{9}+\frac{22041219211}{24412247885}a^{8}+\frac{7278542470}{4882449577}a^{7}+\frac{5865073783}{4882449577}a^{6}+\frac{25691932139}{24412247885}a^{5}-\frac{1169015564}{24412247885}a^{4}-\frac{140013895436}{24412247885}a^{3}-\frac{29357775979}{4882449577}a^{2}-\frac{17830689464}{4882449577}a-\frac{414532274}{567726695}$
| 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: | \( 1326.38565727 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 1326.38565727 \cdot 1}{2\cdot\sqrt{982108001708984375}}\cr\approx \mathstrut & 0.209705353104 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43)
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 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43, 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 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43);
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 - 4*x^13 - 11*x^12 + 9*x^11 + 30*x^10 + 5*x^9 + 7*x^8 - 32*x^7 - 7*x^6 - 78*x^5 - 120*x^4 + 36*x^3 + 154*x^2 + 134*x + 43);
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
$C_5\wr S_3$ (as 15T32):
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 750 |
| The 65 conjugacy class representatives for $C_5\wr S_3$ |
| Character table for $C_5\wr S_3$ |
Intermediate fields
| 3.1.23.1 |
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 15 siblings: | data not computed |
| Degree 30 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 | $15$ | $15$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{5}$ | R | $15$ | $15$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 5.10.16.26 | $x^{10} - 20 x^{9} - 350 x^{8} - 190 x^{5} - 4100 x^{4} - 10975$ | $5$ | $2$ | $16$ | $D_5\times C_5$ | $[2, 2]^{2}$ | |
|
\(23\)
| 23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 23.10.5.2 | $x^{10} + 115 x^{8} + 5296 x^{6} + 36 x^{5} + 120980 x^{4} - 8280 x^{3} + 1383344 x^{2} + 95328 x + 6509876$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |