Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*x - 1)
gp: K = bnfinit(y^15 - y^14 - 6*y^13 + 15*y^12 - 9*y^11 - 13*y^10 + 30*y^9 - 31*y^8 + 24*y^7 - 14*y^6 + 10*y^5 - 18*y^4 + 27*y^3 - 21*y^2 + 8*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*x - 1)
\( x^{15} - x^{14} - 6 x^{13} + 15 x^{12} - 9 x^{11} - 13 x^{10} + 30 x^{9} - 31 x^{8} + 24 x^{7} + \cdots - 1 \)
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: |
\(59463924292145152\)
\(\medspace = 2^{12}\cdot 13^{9}\cdot 37^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.13\) | 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: | $2^{4/5}13^{3/4}37^{2/3}\approx 132.357735064494$ | ||
| Ramified primes: |
\(2\), \(13\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2654}a^{14}-\frac{365}{2654}a^{13}+\frac{77}{1327}a^{12}-\frac{307}{2654}a^{11}+\frac{271}{2654}a^{10}-\frac{459}{2654}a^{9}+\frac{1231}{2654}a^{8}-\frac{458}{1327}a^{7}+\frac{371}{2654}a^{6}+\frac{148}{1327}a^{5}+\frac{540}{1327}a^{4}+\frac{981}{2654}a^{3}-\frac{47}{1327}a^{2}+\frac{510}{1327}a+\frac{144}{1327}$
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: |
$\frac{4997}{2654}a^{14}+\frac{2047}{2654}a^{13}-\frac{15985}{1327}a^{12}+\frac{29127}{2654}a^{11}+\frac{27187}{2654}a^{10}-\frac{31468}{1327}a^{9}+\frac{22890}{1327}a^{8}-\frac{26969}{2654}a^{7}+\frac{19973}{2654}a^{6}+\frac{1744}{1327}a^{5}+\frac{21083}{2654}a^{4}-\frac{25817}{1327}a^{3}+\frac{35869}{2654}a^{2}+\frac{1957}{1327}a-\frac{5967}{2654}$, $\frac{3480}{1327}a^{14}-\frac{4503}{2654}a^{13}-\frac{22747}{1327}a^{12}+\frac{43666}{1327}a^{11}-\frac{18085}{2654}a^{10}-\frac{54019}{1327}a^{9}+\frac{158561}{2654}a^{8}-\frac{67903}{1327}a^{7}+\frac{50335}{1327}a^{6}-\frac{48443}{2654}a^{5}+\frac{44463}{2654}a^{4}-\frac{111123}{2654}a^{3}+\frac{68326}{1327}a^{2}-\frac{38608}{1327}a+\frac{8317}{1327}$, $\frac{940}{1327}a^{14}-\frac{141}{2654}a^{13}-\frac{11709}{2654}a^{12}+\frac{8668}{1327}a^{11}+\frac{1239}{2654}a^{10}-\frac{25583}{2654}a^{9}+\frac{30513}{2654}a^{8}-\frac{24847}{2654}a^{7}+\frac{9028}{1327}a^{6}-\frac{7495}{2654}a^{5}+\frac{6680}{1327}a^{4}-\frac{13395}{1327}a^{3}+\frac{26311}{2654}a^{2}-\frac{4602}{1327}a+\frac{1339}{1327}$, $\frac{12}{1327}a^{14}+\frac{3183}{2654}a^{13}+\frac{521}{1327}a^{12}-\frac{19311}{2654}a^{11}+\frac{9887}{1327}a^{10}+\frac{5108}{1327}a^{9}-\frac{18403}{1327}a^{8}+\frac{37731}{2654}a^{7}-\frac{26925}{2654}a^{6}+\frac{8860}{1327}a^{5}-\frac{1947}{2654}a^{4}+\frac{7791}{1327}a^{3}-\frac{15725}{1327}a^{2}+\frac{14894}{1327}a-\frac{7685}{2654}$, $\frac{2295}{2654}a^{14}-\frac{169}{1327}a^{13}-\frac{16803}{2654}a^{12}+\frac{19977}{2654}a^{11}+\frac{14179}{2654}a^{10}-\frac{39577}{2654}a^{9}+\frac{27829}{2654}a^{8}-\frac{12195}{2654}a^{7}+\frac{7473}{2654}a^{6}+\frac{2602}{1327}a^{5}+\frac{3745}{2654}a^{4}-\frac{16186}{1327}a^{3}+\frac{21803}{2654}a^{2}+\frac{4053}{2654}a-\frac{3924}{1327}$, $\frac{720}{1327}a^{14}-\frac{54}{1327}a^{13}-\frac{10465}{2654}a^{12}+\frac{5877}{1327}a^{11}+\frac{9391}{2654}a^{10}-\frac{22673}{2654}a^{9}+\frac{7846}{1327}a^{8}-\frac{3982}{1327}a^{7}+\frac{3047}{1327}a^{6}+\frac{273}{2654}a^{5}+\frac{3937}{2654}a^{4}-\frac{19193}{2654}a^{3}+\frac{5305}{1327}a^{2}+\frac{2465}{2654}a-\frac{3285}{2654}$, $\frac{601}{2654}a^{14}-\frac{205}{1327}a^{13}-\frac{4317}{2654}a^{12}+\frac{3954}{1327}a^{11}+\frac{977}{2654}a^{10}-\frac{7220}{1327}a^{9}+\frac{12635}{2654}a^{8}-\frac{1896}{1327}a^{7}+\frac{2008}{1327}a^{6}+\frac{39}{1327}a^{5}-\frac{575}{1327}a^{4}-\frac{4448}{1327}a^{3}+\frac{11183}{2654}a^{2}-\frac{27}{1327}a-\frac{3403}{2654}$, $\frac{3480}{1327}a^{14}-\frac{4503}{2654}a^{13}-\frac{22747}{1327}a^{12}+\frac{43666}{1327}a^{11}-\frac{18085}{2654}a^{10}-\frac{54019}{1327}a^{9}+\frac{158561}{2654}a^{8}-\frac{67903}{1327}a^{7}+\frac{50335}{1327}a^{6}-\frac{48443}{2654}a^{5}+\frac{44463}{2654}a^{4}-\frac{111123}{2654}a^{3}+\frac{68326}{1327}a^{2}-\frac{37281}{1327}a+\frac{8317}{1327}$
| 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: | \( 226.669584518 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 226.669584518 \cdot 1}{2\cdot\sqrt{59463924292145152}}\cr\approx \mathstrut & 0.228773473725 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*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 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*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 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*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 - 6*x^13 + 15*x^12 - 9*x^11 - 13*x^10 + 30*x^9 - 31*x^8 + 24*x^7 - 14*x^6 + 10*x^5 - 18*x^4 + 27*x^3 - 21*x^2 + 8*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
$C_3\wr F_5$ (as 15T56):
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 4860 |
| The 63 conjugacy class representatives for $C_3\wr F_5$ |
| Character table for $C_3\wr F_5$ |
Intermediate fields
| 5.1.35152.1 |
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 sibling: | 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 | R | $15$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $15$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.15.12.1 | $x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1$ | $5$ | $3$ | $12$ | $F_5\times C_3$ | $[\ ]_{5}^{12}$ |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.12.9.2 | $x^{12} + 8 x^{10} + 44 x^{9} + 63 x^{8} + 264 x^{7} + 550 x^{6} - 6336 x^{5} + 3843 x^{4} + 4532 x^{3} + 46454 x^{2} + 30668 x + 30982$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
|
\(37\)
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.12.0.1 | $x^{12} + 4 x^{7} + 31 x^{6} + 10 x^{5} + 23 x^{4} + 23 x^{3} + 18 x^{2} + 33 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |