Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*x - 1)
gp: K = bnfinit(y^15 - y^14 - 24*y^13 + 41*y^12 + 165*y^11 - 405*y^10 - 180*y^9 + 963*y^8 - 172*y^7 - 909*y^6 + 338*y^5 + 359*y^4 - 143*y^3 - 47*y^2 + 16*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*x - 1)
\( x^{15} - x^{14} - 24 x^{13} + 41 x^{12} + 165 x^{11} - 405 x^{10} - 180 x^{9} + 963 x^{8} - 172 x^{7} + \cdots - 1 \)
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: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(29766469523035740663169\)
\(\medspace = 19^{6}\cdot 293^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.50\) | 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: | $19^{1/2}293^{1/2}\approx 74.61233142048303$ | ||
| Ramified primes: |
\(19\), \(293\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19}a^{12}+\frac{3}{19}a^{11}+\frac{3}{19}a^{10}+\frac{7}{19}a^{9}+\frac{3}{19}a^{8}+\frac{9}{19}a^{7}+\frac{5}{19}a^{6}+\frac{9}{19}a^{5}-\frac{6}{19}a^{4}+\frac{1}{19}a^{3}+\frac{3}{19}a^{2}+\frac{1}{19}a+\frac{5}{19}$, $\frac{1}{19}a^{13}-\frac{6}{19}a^{11}-\frac{2}{19}a^{10}+\frac{1}{19}a^{9}-\frac{3}{19}a^{7}-\frac{6}{19}a^{6}+\frac{5}{19}a^{5}-\frac{8}{19}a^{2}+\frac{2}{19}a+\frac{4}{19}$, $\frac{1}{1349}a^{14}-\frac{20}{1349}a^{13}+\frac{1}{1349}a^{12}+\frac{519}{1349}a^{11}+\frac{670}{1349}a^{10}-\frac{2}{19}a^{9}+\frac{246}{1349}a^{8}+\frac{478}{1349}a^{7}+\frac{331}{1349}a^{6}+\frac{115}{1349}a^{5}-\frac{498}{1349}a^{4}+\frac{94}{1349}a^{3}-\frac{83}{1349}a^{2}+\frac{465}{1349}a+\frac{411}{1349}$
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: | $14$ | 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: |
$a-1$, $\frac{1247}{1349}a^{14}-\frac{1}{71}a^{13}-\frac{30490}{1349}a^{12}+\frac{22038}{1349}a^{11}+\frac{239871}{1349}a^{10}-\frac{4227}{19}a^{9}-\frac{588406}{1349}a^{8}+\frac{892633}{1349}a^{7}+\frac{578045}{1349}a^{6}-\frac{1042863}{1349}a^{5}-\frac{198556}{1349}a^{4}+\frac{26509}{71}a^{3}-\frac{13615}{1349}a^{2}-\frac{77889}{1349}a+\frac{9695}{1349}$, $\frac{612}{1349}a^{14}+\frac{1037}{1349}a^{13}-\frac{15647}{1349}a^{12}-\frac{13161}{1349}a^{11}+\frac{151245}{1349}a^{10}+\frac{77}{19}a^{9}-\frac{611846}{1349}a^{8}+\frac{337053}{1349}a^{7}+\frac{805149}{1349}a^{6}-\frac{623361}{1349}a^{5}-\frac{343471}{1349}a^{4}+\frac{334002}{1349}a^{3}+\frac{529}{71}a^{2}-\frac{51818}{1349}a+\frac{324}{71}$, $\frac{2040}{1349}a^{14}+\frac{522}{1349}a^{13}-\frac{47092}{1349}a^{12}+\frac{22657}{1349}a^{11}+\frac{338152}{1349}a^{10}-\frac{4770}{19}a^{9}-\frac{653259}{1349}a^{8}+\frac{620759}{1349}a^{7}+\frac{657845}{1349}a^{6}-\frac{362510}{1349}a^{5}-\frac{406811}{1349}a^{4}+\frac{52245}{1349}a^{3}+\frac{5266}{71}a^{2}+\frac{9483}{1349}a-\frac{2768}{1349}$, $\frac{1790}{1349}a^{14}-\frac{797}{1349}a^{13}-\frac{42443}{1349}a^{12}+\frac{49391}{1349}a^{11}+\frac{300511}{1349}a^{10}-394a^{9}-\frac{468034}{1349}a^{8}+\frac{1191592}{1349}a^{7}+\frac{181542}{1349}a^{6}-\frac{1039774}{1349}a^{5}+\frac{66015}{1349}a^{4}+\frac{377639}{1349}a^{3}-\frac{48673}{1349}a^{2}-\frac{48405}{1349}a+\frac{4319}{1349}$, $\frac{1150}{1349}a^{14}-\frac{848}{1349}a^{13}-\frac{27392}{1349}a^{12}+\frac{39713}{1349}a^{11}+\frac{190004}{1349}a^{10}-\frac{5676}{19}a^{9}-\frac{242500}{1349}a^{8}+\frac{48053}{71}a^{7}-\frac{52805}{1349}a^{6}-\frac{800335}{1349}a^{5}+\frac{206951}{1349}a^{4}+\frac{285955}{1349}a^{3}-\frac{83096}{1349}a^{2}-\frac{34954}{1349}a+\frac{7103}{1349}$, $\frac{640}{1349}a^{14}+\frac{51}{1349}a^{13}-\frac{15051}{1349}a^{12}+\frac{9678}{1349}a^{11}+\frac{110507}{1349}a^{10}-\frac{1810}{19}a^{9}-\frac{225534}{1349}a^{8}+\frac{278585}{1349}a^{7}+\frac{234347}{1349}a^{6}-\frac{239439}{1349}a^{5}-\frac{140936}{1349}a^{4}+\frac{91684}{1349}a^{3}+\frac{34423}{1349}a^{2}-\frac{10753}{1349}a-\frac{1435}{1349}$, $\frac{1790}{1349}a^{14}-\frac{797}{1349}a^{13}-\frac{42443}{1349}a^{12}+\frac{49391}{1349}a^{11}+\frac{300511}{1349}a^{10}-394a^{9}-\frac{468034}{1349}a^{8}+\frac{1191592}{1349}a^{7}+\frac{181542}{1349}a^{6}-\frac{1039774}{1349}a^{5}+\frac{66015}{1349}a^{4}+\frac{377639}{1349}a^{3}-\frac{48673}{1349}a^{2}-\frac{48405}{1349}a+\frac{5668}{1349}$, $\frac{603}{1349}a^{14}-\frac{487}{1349}a^{13}-\frac{15798}{1349}a^{12}+\frac{22993}{1349}a^{11}+\frac{132009}{1349}a^{10}-200a^{9}-\frac{328498}{1349}a^{8}+\frac{51227}{71}a^{7}+\frac{79177}{1349}a^{6}-\frac{1226405}{1349}a^{5}+\frac{281054}{1349}a^{4}+\frac{589324}{1349}a^{3}-\frac{155413}{1349}a^{2}-\frac{94627}{1349}a+\frac{17864}{1349}$, $\frac{1150}{1349}a^{14}-\frac{848}{1349}a^{13}-\frac{27392}{1349}a^{12}+\frac{39713}{1349}a^{11}+\frac{190004}{1349}a^{10}-\frac{5676}{19}a^{9}-\frac{242500}{1349}a^{8}+\frac{48053}{71}a^{7}-\frac{52805}{1349}a^{6}-\frac{800335}{1349}a^{5}+\frac{206951}{1349}a^{4}+\frac{285955}{1349}a^{3}-\frac{83096}{1349}a^{2}-\frac{36303}{1349}a+\frac{7103}{1349}$, $a$, $\frac{9}{1349}a^{14}+\frac{1027}{1349}a^{13}-\frac{985}{1349}a^{12}-\frac{23090}{1349}a^{11}+\frac{37057}{1349}a^{10}+\frac{1972}{19}a^{9}-\frac{336669}{1349}a^{8}-\frac{1713}{71}a^{7}+\frac{515528}{1349}a^{6}-\frac{212320}{1349}a^{5}-\frac{12134}{71}a^{4}+\frac{144195}{1349}a^{3}+\frac{17642}{1349}a^{2}-\frac{1196}{71}a+\frac{2208}{1349}$, $\frac{629}{1349}a^{14}+\frac{2472}{1349}a^{13}-\frac{15985}{1349}a^{12}-\frac{47080}{1349}a^{11}+\frac{176906}{1349}a^{10}+\frac{3643}{19}a^{9}-\frac{918999}{1349}a^{8}-\frac{212384}{1349}a^{7}+\frac{1522764}{1349}a^{6}-\frac{66683}{1349}a^{5}-\frac{986535}{1349}a^{4}+\frac{46559}{1349}a^{3}+\frac{228101}{1349}a^{2}+\frac{2450}{1349}a-\frac{306}{71}$, $\frac{1574}{1349}a^{14}-\frac{1944}{1349}a^{13}-\frac{1950}{71}a^{12}+\frac{73252}{1349}a^{11}+\frac{236163}{1349}a^{10}-\frac{9694}{19}a^{9}-\frac{75361}{1349}a^{8}+\frac{1469543}{1349}a^{7}-\frac{700561}{1349}a^{6}-\frac{1095428}{1349}a^{5}+\frac{800939}{1349}a^{4}+\frac{232091}{1349}a^{3}-\frac{223794}{1349}a^{2}+\frac{9059}{1349}a+\frac{140}{71}$
| 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: | \( 1854809.93846 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 1854809.93846 \cdot 1}{2\cdot\sqrt{29766469523035740663169}}\cr\approx \mathstrut & 0.176139066217 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*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 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*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 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*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 - 24*x^13 + 41*x^12 + 165*x^11 - 405*x^10 - 180*x^9 + 963*x^8 - 172*x^7 - 909*x^6 + 338*x^5 + 359*x^4 - 143*x^3 - 47*x^2 + 16*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
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 non-solvable group of order 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| 5.5.30991489.2 |
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 5 sibling: | 5.5.30991489.2 |
| Degree 6 sibling: | 6.6.30991489.1 |
| Degree 10 sibling: | 10.10.960472390437121.1 |
| Degree 12 sibling: | deg 12 |
| Degree 20 sibling: | 20.20.28589872137663846814553020398335096449.1 |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.5.30991489.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{5}$ | ${\href{/padicField/11.3.0.1}{3} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | ${\href{/padicField/29.3.0.1}{3} }^{5}$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | ${\href{/padicField/59.3.0.1}{3} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(293\)
| $\Q_{293}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{293}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{293}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |