Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 6*x^13 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5)
gp: K = bnfinit(y^15 - y^14 + 6*y^13 - 8*y^12 + 25*y^11 - 40*y^10 + 61*y^9 - 74*y^8 + 78*y^7 - 83*y^6 + 82*y^5 - 36*y^4 - 14*y^3 + 35*y^2 - 20*y + 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 + 6*x^13 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 + 6*x^13 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5)
\( x^{15} - x^{14} + 6 x^{13} - 8 x^{12} + 25 x^{11} - 40 x^{10} + 61 x^{9} - 74 x^{8} + 78 x^{7} + \cdots + 5 \)
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: |
\(-1088145913333210475\)
\(\medspace = -\,5^{2}\cdot 17^{4}\cdot 43\cdot 2297^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.94\) | 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: | $5^{2/3}17^{2/3}43^{1/2}2297^{1/2}\approx 6075.667650690129$ | ||
| Ramified primes: |
\(5\), \(17\), \(43\), \(2297\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-98771}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4427274277}a^{14}-\frac{2041988909}{4427274277}a^{13}-\frac{317072872}{4427274277}a^{12}-\frac{1167375872}{4427274277}a^{11}+\frac{1574220660}{4427274277}a^{10}+\frac{20211449}{4427274277}a^{9}+\frac{846853947}{4427274277}a^{8}-\frac{401441157}{4427274277}a^{7}+\frac{1503049984}{4427274277}a^{6}-\frac{993374448}{4427274277}a^{5}-\frac{546385493}{4427274277}a^{4}-\frac{2028981087}{4427274277}a^{3}+\frac{1921235162}{4427274277}a^{2}+\frac{1087271991}{4427274277}a-\frac{468371281}{4427274277}$
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: |
$\frac{481061327}{4427274277}a^{14}+\frac{88244004}{4427274277}a^{13}+\frac{2795885432}{4427274277}a^{12}-\frac{661777790}{4427274277}a^{11}+\frac{9951028268}{4427274277}a^{10}-\frac{7499626450}{4427274277}a^{9}+\frac{15879929735}{4427274277}a^{8}-\frac{14336061245}{4427274277}a^{7}+\frac{12413676701}{4427274277}a^{6}-\frac{17096392169}{4427274277}a^{5}+\frac{10335816810}{4427274277}a^{4}+\frac{6465089228}{4427274277}a^{3}-\frac{6408242940}{4427274277}a^{2}+\frac{10422493055}{4427274277}a-\frac{1469693598}{4427274277}$, $\frac{843522853}{4427274277}a^{14}+\frac{409907170}{4427274277}a^{13}+\frac{4404801806}{4427274277}a^{12}+\frac{124486370}{4427274277}a^{11}+\frac{14385764101}{4427274277}a^{10}-\frac{7580407054}{4427274277}a^{9}+\frac{14835640669}{4427274277}a^{8}-\frac{10795359470}{4427274277}a^{7}+\frac{5412510708}{4427274277}a^{6}-\frac{14472129748}{4427274277}a^{5}+\frac{5873092819}{4427274277}a^{4}+\frac{29339340447}{4427274277}a^{3}-\frac{11653311680}{4427274277}a^{2}-\frac{318994210}{4427274277}a+\frac{6346088336}{4427274277}$, $\frac{506006439}{4427274277}a^{14}-\frac{791986018}{4427274277}a^{13}+\frac{2960304085}{4427274277}a^{12}-\frac{5668884940}{4427274277}a^{11}+\frac{12408919156}{4427274277}a^{10}-\frac{26344401030}{4427274277}a^{9}+\frac{32540924374}{4427274277}a^{8}-\frac{45817175069}{4427274277}a^{7}+\frac{40072725924}{4427274277}a^{6}-\frac{45649379185}{4427274277}a^{5}+\frac{45643249038}{4427274277}a^{4}-\frac{18761155999}{4427274277}a^{3}-\frac{18962407984}{4427274277}a^{2}+\frac{25545875990}{4427274277}a-\frac{11996641901}{4427274277}$, $\frac{1493615410}{4427274277}a^{14}+\frac{101801363}{4427274277}a^{13}+\frac{8443779362}{4427274277}a^{12}-\frac{3055379868}{4427274277}a^{11}+\frac{30626570277}{4427274277}a^{10}-\frac{26027716716}{4427274277}a^{9}+\frac{51273541085}{4427274277}a^{8}-\frac{46319350469}{4427274277}a^{7}+\frac{47931869073}{4427274277}a^{6}-\frac{57288710114}{4427274277}a^{5}+\frac{44010519604}{4427274277}a^{4}+\frac{10917980198}{4427274277}a^{3}-\frac{25330259256}{4427274277}a^{2}+\frac{16400686796}{4427274277}a-\frac{189735708}{4427274277}$, $\frac{323480767}{4427274277}a^{14}-\frac{698591570}{4427274277}a^{13}+\frac{1682279843}{4427274277}a^{12}-\frac{4640845975}{4427274277}a^{11}+\frac{7509024656}{4427274277}a^{10}-\frac{19818423968}{4427274277}a^{9}+\frac{21403808937}{4427274277}a^{8}-\frac{30734130229}{4427274277}a^{7}+\frac{29016497958}{4427274277}a^{6}-\frac{28785050436}{4427274277}a^{5}+\frac{33596573108}{4427274277}a^{4}-\frac{12615749821}{4427274277}a^{3}-\frac{18380514985}{4427274277}a^{2}+\frac{10200120327}{4427274277}a-\frac{5593356392}{4427274277}$, $\frac{629913742}{4427274277}a^{14}-\frac{68387146}{4427274277}a^{13}+\frac{3752240700}{4427274277}a^{12}-\frac{1650788336}{4427274277}a^{11}+\frac{14220128572}{4427274277}a^{10}-\frac{12176616803}{4427274277}a^{9}+\frac{26818314859}{4427274277}a^{8}-\frac{21282677498}{4427274277}a^{7}+\frac{25754942225}{4427274277}a^{6}-\frac{23284903249}{4427274277}a^{5}+\frac{22841824846}{4427274277}a^{4}+\frac{1331099366}{4427274277}a^{3}-\frac{7475600307}{4427274277}a^{2}+\frac{9893002865}{4427274277}a-\frac{1568054826}{4427274277}$, $\frac{1736308951}{4427274277}a^{14}-\frac{379166930}{4427274277}a^{13}+\frac{9273794475}{4427274277}a^{12}-\frac{6591864512}{4427274277}a^{11}+\frac{33637615387}{4427274277}a^{10}-\frac{40863642234}{4427274277}a^{9}+\frac{57501926559}{4427274277}a^{8}-\frac{66638596504}{4427274277}a^{7}+\frac{57260706638}{4427274277}a^{6}-\frac{72368392188}{4427274277}a^{5}+\frac{63252962927}{4427274277}a^{4}+\frac{17050206264}{4427274277}a^{3}-\frac{36448480055}{4427274277}a^{2}+\frac{18032659059}{4427274277}a-\frac{1652574691}{4427274277}$
| 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: | \( 786.568254332 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 786.568254332 \cdot 1}{2\cdot\sqrt{1088145913333210475}}\cr\approx \mathstrut & 0.291508971555 \end{aligned}\]
# 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 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5)
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 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5, 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 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5);
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 - 8*x^12 + 25*x^11 - 40*x^10 + 61*x^9 - 74*x^8 + 78*x^7 - 83*x^6 + 82*x^5 - 36*x^4 - 14*x^3 + 35*x^2 - 20*x + 5);
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
$S_3^5.S_5$ (as 15T93):
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 933120 |
| The 108 conjugacy class representatives for $S_3^5.S_5$ |
| Character table for $S_3^5.S_5$ |
Intermediate fields
| 5.1.2297.1 |
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 |
| Degree 45 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $15$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | R | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.6.4.2 | $x^{6} + 204 x^{3} - 7225$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(43\)
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.12.0.1 | $x^{12} + 34 x^{7} + 27 x^{6} + 16 x^{5} + 17 x^{4} + 6 x^{3} + 23 x^{2} + 38 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(2297\)
| $\Q_{2297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |