Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1)
gp: K = bnfinit(y^15 - y^14 - y^13 + y^12 - 7*y^11 - 2*y^10 + 18*y^9 + 12*y^8 - 17*y^7 + 7*y^6 - 11*y^5 - 21*y^4 + 22*y^3 - 11*y^2 - 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*x - 1)
\( x^{15} - x^{14} - x^{13} + x^{12} - 7 x^{11} - 2 x^{10} + 18 x^{9} + 12 x^{8} - 17 x^{7} + 7 x^{6} + \cdots - 1 \)
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: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(21643467887730481\)
\(\medspace = 13^{2}\cdot 47^{6}\cdot 109^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.28\) | 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: | $13^{2/3}47^{1/2}109^{2/3}\approx 864.8925624711881$ | ||
| Ramified primes: |
\(13\), \(47\), \(109\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4823316529}a^{14}-\frac{240451914}{4823316529}a^{13}-\frac{1467210693}{4823316529}a^{12}+\frac{2044776038}{4823316529}a^{11}+\frac{2380646120}{4823316529}a^{10}-\frac{2290615831}{4823316529}a^{9}-\frac{1561717558}{4823316529}a^{8}+\frac{2080171999}{4823316529}a^{7}-\frac{2386498101}{4823316529}a^{6}+\frac{1715950732}{4823316529}a^{5}+\frac{1834453739}{4823316529}a^{4}+\frac{64937028}{4823316529}a^{3}+\frac{593205418}{4823316529}a^{2}+\frac{970975456}{4823316529}a+\frac{820723294}{4823316529}$
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: |
$\frac{438282190}{4823316529}a^{14}-\frac{666156758}{4823316529}a^{13}-\frac{180199038}{4823316529}a^{12}+\frac{937506764}{4823316529}a^{11}-\frac{3215866687}{4823316529}a^{10}+\frac{764978875}{4823316529}a^{9}+\frac{8142107838}{4823316529}a^{8}-\frac{819825977}{4823316529}a^{7}-\frac{12445039424}{4823316529}a^{6}+\frac{7101345595}{4823316529}a^{5}-\frac{2075689344}{4823316529}a^{4}-\frac{3242761291}{4823316529}a^{3}+\frac{18275305195}{4823316529}a^{2}-\frac{11471590121}{4823316529}a-\frac{1755151205}{4823316529}$, $\frac{279318677}{4823316529}a^{14}-\frac{570733139}{4823316529}a^{13}-\frac{97673519}{4823316529}a^{12}+\frac{286289673}{4823316529}a^{11}-\frac{2256835527}{4823316529}a^{10}+\frac{1369023982}{4823316529}a^{9}+\frac{6081306034}{4823316529}a^{8}+\frac{1059190063}{4823316529}a^{7}-\frac{6065350387}{4823316529}a^{6}+\frac{5660375798}{4823316529}a^{5}-\frac{6997088601}{4823316529}a^{4}-\frac{5291197834}{4823316529}a^{3}+\frac{4910463282}{4823316529}a^{2}-\frac{11751794763}{4823316529}a+\frac{4235471514}{4823316529}$, $\frac{38028855}{4823316529}a^{14}+\frac{30131136}{4823316529}a^{13}-\frac{41830710}{4823316529}a^{12}+\frac{95826109}{4823316529}a^{11}+\frac{13051036}{4823316529}a^{10}-\frac{448791598}{4823316529}a^{9}+\frac{291200427}{4823316529}a^{8}+\frac{903950089}{4823316529}a^{7}-\frac{1148631260}{4823316529}a^{6}-\frac{1349674114}{4823316529}a^{5}+\frac{1574584533}{4823316529}a^{4}+\frac{638893288}{4823316529}a^{3}-\frac{3508838763}{4823316529}a^{2}+\frac{1850995278}{4823316529}a+\frac{1978357973}{4823316529}$, $\frac{566139539}{4823316529}a^{14}-\frac{325368897}{4823316529}a^{13}-\frac{420696872}{4823316529}a^{12}+\frac{578463500}{4823316529}a^{11}-\frac{3451205678}{4823316529}a^{10}-\frac{2747557933}{4823316529}a^{9}+\frac{7365442564}{4823316529}a^{8}+\frac{6303751499}{4823316529}a^{7}-\frac{9186881148}{4823316529}a^{6}+\frac{2757048777}{4823316529}a^{5}-\frac{662352361}{4823316529}a^{4}-\frac{5601051546}{4823316529}a^{3}+\frac{12324635935}{4823316529}a^{2}+\frac{209859584}{4823316529}a-\frac{2375337177}{4823316529}$, $\frac{469621379}{4823316529}a^{14}+\frac{58489305}{4823316529}a^{13}-\frac{825121412}{4823316529}a^{12}+\frac{90755217}{4823316529}a^{11}-\frac{2804712262}{4823316529}a^{10}-\frac{4403499472}{4823316529}a^{9}+\frac{6154418487}{4823316529}a^{8}+\frac{12709698685}{4823316529}a^{7}-\frac{2583762033}{4823316529}a^{6}-\frac{4750900235}{4823316529}a^{5}-\frac{1059112278}{4823316529}a^{4}-\frac{12098985853}{4823316529}a^{3}+\frac{4091725504}{4823316529}a^{2}+\frac{10667639529}{4823316529}a-\frac{2580378452}{4823316529}$, $\frac{1205387739}{4823316529}a^{14}-\frac{816773200}{4823316529}a^{13}-\frac{1373326882}{4823316529}a^{12}+\frac{706262859}{4823316529}a^{11}-\frac{7858948337}{4823316529}a^{10}-\frac{4862149836}{4823316529}a^{9}+\frac{19365056971}{4823316529}a^{8}+\frac{20389332875}{4823316529}a^{7}-\frac{14912893927}{4823316529}a^{6}+\frac{1147428040}{4823316529}a^{5}-\frac{12087988177}{4823316529}a^{4}-\frac{21912285059}{4823316529}a^{3}+\frac{18575332517}{4823316529}a^{2}-\frac{6222009777}{4823316529}a+\frac{700795973}{4823316529}$, $\frac{667132927}{4823316529}a^{14}-\frac{753110120}{4823316529}a^{13}-\frac{1017644411}{4823316529}a^{12}+\frac{516490886}{4823316529}a^{11}-\frac{4405300094}{4823316529}a^{10}-\frac{813451459}{4823316529}a^{9}+\frac{14786449294}{4823316529}a^{8}+\frac{11452381693}{4823316529}a^{7}-\frac{13822873461}{4823316529}a^{6}-\frac{3214737208}{4823316529}a^{5}-\frac{9422680395}{4823316529}a^{4}-\frac{15211313221}{4823316529}a^{3}+\frac{15929636186}{4823316529}a^{2}-\frac{4384209022}{4823316529}a-\frac{4901669196}{4823316529}$, $\frac{648967434}{4823316529}a^{14}-\frac{1114737512}{4823316529}a^{13}-\frac{747311498}{4823316529}a^{12}+\frac{1124779681}{4823316529}a^{11}-\frac{4823766233}{4823316529}a^{10}+\frac{1779201762}{4823316529}a^{9}+\frac{15799224682}{4823316529}a^{8}+\frac{4012111589}{4823316529}a^{7}-\frac{20004131378}{4823316529}a^{6}+\frac{3648655910}{4823316529}a^{5}-\frac{7466582441}{4823316529}a^{4}-\frac{13931973140}{4823316529}a^{3}+\frac{24631797650}{4823316529}a^{2}-\frac{10717944600}{4823316529}a-\frac{879206783}{4823316529}$
| 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: | \( 116.633322592 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 116.633322592 \cdot 1}{2\cdot\sqrt{21643467887730481}}\cr\approx \mathstrut & 0.195118489473 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*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^15 - x^14 - x^13 + x^12 - 7*x^11 - 2*x^10 + 18*x^9 + 12*x^8 - 17*x^7 + 7*x^6 - 11*x^5 - 21*x^4 + 22*x^3 - 11*x^2 - 2*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
$C_3\wr D_5$ (as 15T46):
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 2430 |
| The 72 conjugacy class representatives for $C_3\wr D_5$ |
| Character table for $C_3\wr D_5$ |
Intermediate fields
| 5.1.2209.1 |
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 |
| Degree 45 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 | $15$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $15$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | $15$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $15$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | R | $15$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(47\)
| 47.3.0.1 | $x^{3} + 3 x + 42$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 47.6.3.2 | $x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 47.6.3.2 | $x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(109\)
| 109.3.2.2 | $x^{3} + 327$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 109.6.0.1 | $x^{6} + 107 x^{3} + 102 x^{2} + 66 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 109.6.0.1 | $x^{6} + 107 x^{3} + 102 x^{2} + 66 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |