Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*x + 1)
gp: K = bnfinit(y^15 - 5*y^14 + 14*y^13 - 23*y^12 + 19*y^11 + 7*y^10 - 41*y^9 + 47*y^8 - 10*y^7 - 39*y^6 + 50*y^5 - 18*y^4 - 16*y^3 + 21*y^2 - 9*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*x + 1)
\( x^{15} - 5 x^{14} + 14 x^{13} - 23 x^{12} + 19 x^{11} + 7 x^{10} - 41 x^{9} + 47 x^{8} - 10 x^{7} + \cdots + 1 \)
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: |
\(19664802801826816\)
\(\medspace = 2^{10}\cdot 79^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.20\) | 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: | $2\cdot 79^{1/2}\approx 17.776388834631177$ | ||
| Ramified primes: |
\(2\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{79}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17}a^{14}-\frac{8}{17}a^{13}+\frac{4}{17}a^{12}-\frac{1}{17}a^{11}+\frac{5}{17}a^{10}-\frac{8}{17}a^{9}-\frac{4}{17}a^{7}+\frac{2}{17}a^{6}+\frac{6}{17}a^{5}-\frac{2}{17}a^{4}+\frac{5}{17}a^{3}+\frac{3}{17}a^{2}-\frac{5}{17}a+\frac{6}{17}$
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: |
$\frac{13}{17}a^{14}-\frac{53}{17}a^{13}+\frac{120}{17}a^{12}-\frac{132}{17}a^{11}-\frac{3}{17}a^{10}+\frac{236}{17}a^{9}-19a^{8}+\frac{67}{17}a^{7}+\frac{298}{17}a^{6}-\frac{364}{17}a^{5}+\frac{59}{17}a^{4}+\frac{201}{17}a^{3}-\frac{148}{17}a^{2}-\frac{14}{17}a+\frac{27}{17}$, $\frac{14}{17}a^{14}-\frac{61}{17}a^{13}+\frac{158}{17}a^{12}-\frac{235}{17}a^{11}+\frac{155}{17}a^{10}+\frac{126}{17}a^{9}-26a^{8}+\frac{403}{17}a^{7}+\frac{11}{17}a^{6}-\frac{443}{17}a^{5}+\frac{448}{17}a^{4}-\frac{66}{17}a^{3}-\frac{196}{17}a^{2}+\frac{185}{17}a-\frac{35}{17}$, $\frac{1}{17}a^{14}-\frac{8}{17}a^{13}+\frac{21}{17}a^{12}-\frac{35}{17}a^{11}+\frac{22}{17}a^{10}+\frac{26}{17}a^{9}-4a^{8}+\frac{64}{17}a^{7}+\frac{19}{17}a^{6}-\frac{62}{17}a^{5}+\frac{49}{17}a^{4}+\frac{5}{17}a^{3}-\frac{31}{17}a^{2}+\frac{12}{17}a-\frac{11}{17}$, $\frac{5}{17}a^{14}-\frac{6}{17}a^{13}-\frac{14}{17}a^{12}+\frac{80}{17}a^{11}-\frac{145}{17}a^{10}+\frac{79}{17}a^{9}+8a^{8}-\frac{326}{17}a^{7}+\frac{197}{17}a^{6}+\frac{166}{17}a^{5}-\frac{350}{17}a^{4}+\frac{161}{17}a^{3}+\frac{117}{17}a^{2}-\frac{144}{17}a+\frac{30}{17}$, $\frac{23}{17}a^{14}-\frac{99}{17}a^{13}+\frac{245}{17}a^{12}-\frac{329}{17}a^{11}+\frac{149}{17}a^{10}+\frac{309}{17}a^{9}-40a^{8}+\frac{452}{17}a^{7}+\frac{216}{17}a^{6}-\frac{678}{17}a^{5}+\frac{447}{17}a^{4}+\frac{64}{17}a^{3}-\frac{271}{17}a^{2}+\frac{157}{17}a-\frac{32}{17}$, $\frac{7}{17}a^{14}-\frac{22}{17}a^{13}+\frac{45}{17}a^{12}-\frac{41}{17}a^{11}-\frac{16}{17}a^{10}+\frac{97}{17}a^{9}-8a^{8}+\frac{40}{17}a^{7}+\frac{99}{17}a^{6}-\frac{162}{17}a^{5}+\frac{71}{17}a^{4}+\frac{69}{17}a^{3}-\frac{98}{17}a^{2}+\frac{33}{17}a+\frac{8}{17}$, $\frac{14}{17}a^{14}-\frac{61}{17}a^{13}+\frac{158}{17}a^{12}-\frac{235}{17}a^{11}+\frac{155}{17}a^{10}+\frac{126}{17}a^{9}-26a^{8}+\frac{403}{17}a^{7}+\frac{11}{17}a^{6}-\frac{443}{17}a^{5}+\frac{448}{17}a^{4}-\frac{66}{17}a^{3}-\frac{196}{17}a^{2}+\frac{185}{17}a-\frac{52}{17}$, $\frac{13}{17}a^{14}-\frac{53}{17}a^{13}+\frac{120}{17}a^{12}-\frac{132}{17}a^{11}-\frac{3}{17}a^{10}+\frac{236}{17}a^{9}-19a^{8}+\frac{67}{17}a^{7}+\frac{298}{17}a^{6}-\frac{364}{17}a^{5}+\frac{59}{17}a^{4}+\frac{201}{17}a^{3}-\frac{148}{17}a^{2}+\frac{3}{17}a+\frac{27}{17}$
| 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: | \( 103.698212623 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 103.698212623 \cdot 1}{2\cdot\sqrt{19664802801826816}}\cr\approx \mathstrut & 0.181997605641 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*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 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*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 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*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 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*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
| 3.3.316.1, 5.1.6241.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $15$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.1.1 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |