Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*x - 1)
gp: K = bnfinit(y^15 - 6*y^14 + 19*y^13 - 39*y^12 + 58*y^11 - 67*y^10 + 68*y^9 - 69*y^8 + 65*y^7 - 52*y^6 + 33*y^5 - 20*y^4 + 12*y^3 - 4*y^2 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*x - 1)
\( x^{15} - 6 x^{14} + 19 x^{13} - 39 x^{12} + 58 x^{11} - 67 x^{10} + 68 x^{9} - 69 x^{8} + 65 x^{7} + \cdots - 1 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-96318312824155136\)
\(\medspace = -\,2^{10}\cdot 11^{7}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.56\) | 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: | $2^{2/3}11^{1/2}13^{1/2}\approx 18.982555683309183$ | ||
| Ramified primes: |
\(2\), \(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11}a^{13}-\frac{2}{11}a^{12}-\frac{4}{11}a^{11}-\frac{3}{11}a^{10}-\frac{4}{11}a^{9}-\frac{5}{11}a^{8}-\frac{2}{11}a^{7}-\frac{2}{11}a^{6}-\frac{1}{11}a^{5}-\frac{4}{11}a^{4}-\frac{1}{11}a^{3}+\frac{3}{11}a^{2}-\frac{5}{11}a-\frac{3}{11}$, $\frac{1}{319}a^{14}-\frac{10}{319}a^{13}+\frac{1}{319}a^{12}+\frac{73}{319}a^{11}-\frac{2}{319}a^{10}+\frac{115}{319}a^{9}+\frac{159}{319}a^{8}-\frac{96}{319}a^{7}-\frac{73}{319}a^{6}+\frac{37}{319}a^{5}-\frac{57}{319}a^{4}+\frac{11}{29}a^{3}-\frac{95}{319}a^{2}-\frac{117}{319}a+\frac{123}{319}$
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: | $7$ | 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: |
$\frac{163}{319}a^{14}-\frac{615}{319}a^{13}+\frac{1323}{319}a^{12}-\frac{1412}{319}a^{11}+\frac{776}{319}a^{10}+\frac{30}{29}a^{9}+\frac{107}{319}a^{8}-\frac{771}{319}a^{7}+\frac{426}{319}a^{6}+\frac{21}{29}a^{5}-\frac{591}{319}a^{4}-\frac{751}{319}a^{3}+\frac{1}{319}a^{2}-\frac{540}{319}a+\frac{97}{319}$, $\frac{195}{319}a^{14}-\frac{848}{319}a^{13}+\frac{2138}{319}a^{12}-\frac{3252}{319}a^{11}+\frac{331}{29}a^{10}-\frac{3037}{319}a^{9}+\frac{3165}{319}a^{8}-\frac{3379}{319}a^{7}+\frac{2382}{319}a^{6}-\frac{1224}{319}a^{5}+\frac{108}{319}a^{4}-\frac{1113}{319}a^{3}+\frac{93}{319}a^{2}+\frac{6}{29}a+\frac{263}{319}$, $\frac{33}{29}a^{14}-\frac{1774}{319}a^{13}+\frac{4945}{319}a^{12}-\frac{8678}{319}a^{11}+\frac{11251}{319}a^{10}-\frac{10977}{319}a^{9}+\frac{10476}{319}a^{8}-\frac{10169}{319}a^{7}+\frac{8707}{319}a^{6}-\frac{5970}{319}a^{5}+\frac{2509}{319}a^{4}-\frac{1955}{319}a^{3}-\frac{207}{319}a^{2}-\frac{392}{319}a+\frac{163}{319}$, $\frac{13}{319}a^{14}-\frac{43}{319}a^{13}+\frac{158}{319}a^{12}-\frac{356}{319}a^{11}+\frac{670}{319}a^{10}-\frac{767}{319}a^{9}+\frac{675}{319}a^{8}-\frac{146}{319}a^{7}-\frac{166}{319}a^{6}+\frac{394}{319}a^{5}-\frac{70}{29}a^{4}+\frac{529}{319}a^{3}-\frac{655}{319}a^{2}-\frac{42}{319}a-\frac{257}{319}$, $\frac{183}{319}a^{14}-\frac{1134}{319}a^{13}+\frac{3576}{319}a^{12}-\frac{7289}{319}a^{11}+\frac{10625}{319}a^{10}-\frac{12044}{319}a^{9}+\frac{11900}{319}a^{8}-\frac{11942}{319}a^{7}+\frac{979}{29}a^{6}-\frac{7961}{319}a^{5}+\frac{4649}{319}a^{4}-\frac{2478}{319}a^{3}+\frac{1291}{319}a^{2}+\frac{310}{319}a+\frac{324}{319}$, $a-1$, $\frac{60}{319}a^{14}-\frac{426}{319}a^{13}+\frac{1307}{319}a^{12}-\frac{2377}{319}a^{11}+\frac{2548}{319}a^{10}-\frac{132}{29}a^{9}+\frac{57}{319}a^{8}-\frac{47}{319}a^{7}+\frac{57}{319}a^{6}+\frac{128}{29}a^{5}-\frac{2521}{319}a^{4}+\frac{1982}{319}a^{3}-\frac{393}{319}a^{2}+\frac{723}{319}a-\frac{479}{319}$
| 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: | \( 226.323474369 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 226.323474369 \cdot 1}{2\cdot\sqrt{96318312824155136}}\cr\approx \mathstrut & 0.281925426892 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*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 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*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 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*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 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*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
$S_3\times D_5$ (as 15T7):
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 60 |
| The 12 conjugacy class representatives for $D_5\times S_3$ |
| Character table for $D_5\times S_3$ |
Intermediate fields
| 3.1.44.1, 5.1.20449.1 |
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 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 | R | $15$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | R | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $15$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.15.10.1 | $x^{15} + 13 x^{12} + 3 x^{10} + 13 x^{9} - 174 x^{7} - 45 x^{6} + 3 x^{5} + 513 x^{4} + 63 x^{2} - 216 x + 109$ | $3$ | $5$ | $10$ | $S_3 \times C_5$ | $[\ ]_{3}^{10}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |