Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 1)
gp: K = bnfinit(y^15 - 2*y^14 + 2*y^13 - 2*y^12 + 2*y^11 - 4*y^10 + 3*y^9 + 2*y^8 + y^7 - 5*y^6 + 3*y^5 + 3*y^4 + 6*y^3 + 2*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 1)
\( x^{15} - 2 x^{14} + 2 x^{13} - 2 x^{12} + 2 x^{11} - 4 x^{10} + 3 x^{9} + 2 x^{8} + x^{7} - 5 x^{6} + \cdots - 1 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-69378727128301847\)
\(\medspace = -\,23^{5}\cdot 47^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.27\) | 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: | $23^{1/2}47^{1/2}\approx 32.87856444554719$ | ||
| Ramified primes: |
\(23\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5995}a^{14}-\frac{258}{5995}a^{13}+\frac{21}{1199}a^{12}-\frac{2902}{5995}a^{11}-\frac{466}{5995}a^{10}-\frac{608}{5995}a^{9}-\frac{219}{5995}a^{8}+\frac{2111}{5995}a^{7}-\frac{173}{1199}a^{6}-\frac{76}{1199}a^{5}+\frac{1363}{5995}a^{4}-\frac{243}{1199}a^{3}-\frac{694}{5995}a^{2}-\frac{2184}{5995}a+\frac{1569}{5995}$
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: |
$\frac{1982}{5995}a^{14}-\frac{596}{1199}a^{13}+\frac{683}{5995}a^{12}+\frac{2237}{5995}a^{11}-\frac{3979}{5995}a^{10}-\frac{61}{5995}a^{9}-\frac{3617}{5995}a^{8}+\frac{15079}{5995}a^{7}-\frac{7054}{5995}a^{6}-\frac{6183}{5995}a^{5}+\frac{3716}{5995}a^{4}+\frac{15049}{5995}a^{3}+\frac{4541}{5995}a^{2}+\frac{9294}{5995}a-\frac{1647}{5995}$, $\frac{1006}{5995}a^{14}-\frac{2962}{5995}a^{13}+\frac{2516}{5995}a^{12}+\frac{153}{5995}a^{11}-\frac{477}{1199}a^{10}+\frac{1041}{5995}a^{9}+\frac{302}{5995}a^{8}+\frac{7431}{5995}a^{7}-\frac{9308}{5995}a^{6}-\frac{919}{1199}a^{5}+\frac{9114}{5995}a^{4}+\frac{1889}{5995}a^{3}-\frac{3943}{5995}a^{2}-\frac{347}{1199}a-\frac{1093}{1199}$, $\frac{1044}{5995}a^{14}-\frac{395}{1199}a^{13}+\frac{511}{5995}a^{12}+\frac{1384}{5995}a^{11}-\frac{2108}{5995}a^{10}+\frac{718}{5995}a^{9}-\frac{3224}{5995}a^{8}+\frac{10913}{5995}a^{7}-\frac{6208}{5995}a^{6}-\frac{5846}{5995}a^{5}+\frac{2157}{5995}a^{4}+\frac{10873}{5995}a^{3}+\frac{3257}{5995}a^{2}-\frac{797}{5995}a-\frac{4594}{5995}$, $\frac{2166}{5995}a^{14}-\frac{3691}{5995}a^{13}+\frac{2018}{5995}a^{12}-\frac{574}{5995}a^{11}+\frac{202}{5995}a^{10}-\frac{5222}{5995}a^{9}+\frac{2848}{5995}a^{8}+\frac{9032}{5995}a^{7}+\frac{1646}{5995}a^{6}-\frac{14954}{5995}a^{5}+\frac{9912}{5995}a^{4}+\frac{8508}{5995}a^{3}+\frac{12332}{5995}a^{2}+\frac{9103}{5995}a+\frac{488}{5995}$, $\frac{575}{1199}a^{14}-\frac{873}{1199}a^{13}-\frac{273}{5995}a^{12}+\frac{6586}{5995}a^{11}-\frac{11258}{5995}a^{10}+\frac{7336}{5995}a^{9}-\frac{2428}{1199}a^{8}+\frac{29762}{5995}a^{7}-\frac{15736}{5995}a^{6}-\frac{9803}{5995}a^{5}+\frac{5089}{5995}a^{4}+\frac{3989}{1199}a^{3}+\frac{3483}{5995}a^{2}+\frac{18148}{5995}a-\frac{8156}{5995}$, $\frac{387}{1199}a^{14}-\frac{6441}{5995}a^{13}+\frac{8937}{5995}a^{12}-\frac{8846}{5995}a^{11}+\frac{7132}{5995}a^{10}-\frac{1491}{1199}a^{9}+\frac{9074}{5995}a^{8}+\frac{4588}{5995}a^{7}-\frac{11961}{5995}a^{6}-\frac{1512}{5995}a^{5}+\frac{1120}{1199}a^{4}-\frac{2184}{5995}a^{3}+\frac{4786}{5995}a^{2}-\frac{3162}{5995}a-\frac{690}{1199}$, $\frac{174}{545}a^{14}-\frac{84}{109}a^{13}+\frac{612}{545}a^{12}-\frac{823}{545}a^{11}+\frac{993}{545}a^{10}-\frac{1479}{545}a^{9}+\frac{1461}{545}a^{8}-\frac{561}{545}a^{7}+\frac{564}{545}a^{6}-\frac{144}{109}a^{5}+\frac{414}{545}a^{4}+\frac{268}{545}a^{3}+\frac{1106}{545}a^{2}+\frac{67}{545}a+\frac{288}{545}$
| 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: | \( 185.949282778 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 185.949282778 \cdot 1}{2\cdot\sqrt{69378727128301847}}\cr\approx \mathstrut & 0.272923179603 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 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 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 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 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 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 - 2*x^14 + 2*x^13 - 2*x^12 + 2*x^11 - 4*x^10 + 3*x^9 + 2*x^8 + x^7 - 5*x^6 + 3*x^5 + 3*x^4 + 6*x^3 + 2*x^2 - 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
| 3.1.23.1, 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 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 | $15$ | $15$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | R | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(47\)
| 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 | $x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |