Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1)
gp: K = bnfinit(y^15 - 4*y^14 + 7*y^13 - 17*y^12 + 51*y^11 - 87*y^10 - 7*y^9 + 172*y^8 - 100*y^7 - 69*y^6 - 17*y^5 + 62*y^4 + 50*y^3 - 22*y^2 - 18*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*x - 1)
\( x^{15} - 4 x^{14} + 7 x^{13} - 17 x^{12} + 51 x^{11} - 87 x^{10} - 7 x^{9} + 172 x^{8} - 100 x^{7} + \cdots - 1 \)
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: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-24417546297445042591\)
\(\medspace = -\,31^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.61\) | 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: | $31^{9/10}\approx 21.990014941549536$ | ||
| Ramified primes: |
\(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{67}a^{13}-\frac{30}{67}a^{12}+\frac{15}{67}a^{11}-\frac{27}{67}a^{10}+\frac{27}{67}a^{9}+\frac{22}{67}a^{8}+\frac{17}{67}a^{7}+\frac{32}{67}a^{6}+\frac{14}{67}a^{5}-\frac{12}{67}a^{4}+\frac{6}{67}a^{3}-\frac{9}{67}a^{2}+\frac{7}{67}a-\frac{23}{67}$, $\frac{1}{2328257544133}a^{14}+\frac{16975831322}{2328257544133}a^{13}+\frac{935820820203}{2328257544133}a^{12}-\frac{434972844569}{2328257544133}a^{11}-\frac{814302805997}{2328257544133}a^{10}-\frac{211932806836}{2328257544133}a^{9}+\frac{12371814752}{2328257544133}a^{8}-\frac{357482693880}{2328257544133}a^{7}-\frac{6663952029}{2328257544133}a^{6}-\frac{683853589830}{2328257544133}a^{5}-\frac{1156621051070}{2328257544133}a^{4}+\frac{132213504029}{2328257544133}a^{3}-\frac{1154630429440}{2328257544133}a^{2}-\frac{892730603990}{2328257544133}a+\frac{1076347464613}{2328257544133}$
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: | $9$ | 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: |
$\frac{243059409}{2328257544133}a^{14}-\frac{58795611972}{2328257544133}a^{13}+\frac{283835217917}{2328257544133}a^{12}-\frac{671174577289}{2328257544133}a^{11}+\frac{1545391784548}{2328257544133}a^{10}-\frac{4167500959675}{2328257544133}a^{9}+\frac{8618200703888}{2328257544133}a^{8}-\frac{6698796274221}{2328257544133}a^{7}-\frac{6040944747651}{2328257544133}a^{6}+\frac{14805483642160}{2328257544133}a^{5}-\frac{5586924398371}{2328257544133}a^{4}-\frac{724165140729}{2328257544133}a^{3}-\frac{5814004335306}{2328257544133}a^{2}+\frac{4159657329275}{2328257544133}a+\frac{3685669397498}{2328257544133}$, $\frac{267585097919}{2328257544133}a^{14}-\frac{1253520182462}{2328257544133}a^{13}+\frac{2641390360838}{2328257544133}a^{12}-\frac{6138719447285}{2328257544133}a^{11}+\frac{17569473744530}{2328257544133}a^{10}-\frac{34144131166079}{2328257544133}a^{9}+\frac{18485848976355}{2328257544133}a^{8}+\frac{36794259919607}{2328257544133}a^{7}-\frac{47010188618512}{2328257544133}a^{6}+\frac{9288455272432}{2328257544133}a^{5}-\frac{14348444844804}{2328257544133}a^{4}+\frac{23900361256751}{2328257544133}a^{3}+\frac{6093122570911}{2328257544133}a^{2}-\frac{4109472049064}{2328257544133}a-\frac{2536176204826}{2328257544133}$, $\frac{107616096499}{2328257544133}a^{14}-\frac{279921774352}{2328257544133}a^{13}+\frac{273579756429}{2328257544133}a^{12}-\frac{1174327988520}{2328257544133}a^{11}+\frac{3541136186138}{2328257544133}a^{10}-\frac{3439217257266}{2328257544133}a^{9}-\frac{8823909868513}{2328257544133}a^{8}+\frac{10130838537873}{2328257544133}a^{7}+\frac{10141573761668}{2328257544133}a^{6}-\frac{5961882724531}{2328257544133}a^{5}-\frac{18352065081316}{2328257544133}a^{4}-\frac{664268199043}{2328257544133}a^{3}+\frac{11897335245225}{2328257544133}a^{2}+\frac{3226043066405}{2328257544133}a-\frac{827874003385}{2328257544133}$, $\frac{15534851802}{2328257544133}a^{14}-\frac{310420367835}{2328257544133}a^{13}+\frac{846059759719}{2328257544133}a^{12}-\frac{1224851620432}{2328257544133}a^{11}+\frac{4012431435855}{2328257544133}a^{10}-\frac{10833583530070}{2328257544133}a^{9}+\frac{11924086802913}{2328257544133}a^{8}+\frac{16551040214255}{2328257544133}a^{7}-\frac{28146605200127}{2328257544133}a^{6}-\frac{9744470481690}{2328257544133}a^{5}+\frac{11980428858427}{2328257544133}a^{4}+\frac{27406648660638}{2328257544133}a^{3}-\frac{1611692855579}{2328257544133}a^{2}-\frac{13921249553043}{2328257544133}a-\frac{3308306735870}{2328257544133}$, $\frac{221069717989}{2328257544133}a^{14}-\frac{812803982821}{2328257544133}a^{13}+\frac{1441695630178}{2328257544133}a^{12}-\frac{3830724914016}{2328257544133}a^{11}+\frac{10919146009939}{2328257544133}a^{10}-\frac{18122667063096}{2328257544133}a^{9}-\frac{453458022259}{2328257544133}a^{8}+\frac{27069382635229}{2328257544133}a^{7}-\frac{17519724693137}{2328257544133}a^{6}+\frac{69067802221}{2328257544133}a^{5}-\frac{13704436043447}{2328257544133}a^{4}+\frac{4042596563740}{2328257544133}a^{3}+\frac{4776557664914}{2328257544133}a^{2}+\frac{1488718191345}{2328257544133}a+\frac{3046632114016}{2328257544133}$, $\frac{106584952006}{2328257544133}a^{14}-\frac{459656461286}{2328257544133}a^{13}+\frac{1012465523189}{2328257544133}a^{12}-\frac{2613992273846}{2328257544133}a^{11}+\frac{7131863600067}{2328257544133}a^{10}-\frac{13756623825798}{2328257544133}a^{9}+\frac{10104727394380}{2328257544133}a^{8}+\frac{3923788363950}{2328257544133}a^{7}-\frac{10441282622150}{2328257544133}a^{6}+\frac{11823397836943}{2328257544133}a^{5}-\frac{15073170859941}{2328257544133}a^{4}+\frac{9596079935112}{2328257544133}a^{3}-\frac{2226004000257}{2328257544133}a^{2}-\frac{1474073828694}{2328257544133}a+\frac{942994534295}{2328257544133}$, $\frac{1151156254397}{2328257544133}a^{14}-\frac{4823448537837}{2328257544133}a^{13}+\frac{9134091304009}{2328257544133}a^{12}-\frac{21865623729734}{2328257544133}a^{11}+\frac{63766691902375}{2328257544133}a^{10}-\frac{114734559068094}{2328257544133}a^{9}+\frac{20949315679253}{2328257544133}a^{8}+\frac{182877151681740}{2328257544133}a^{7}-\frac{154002735802108}{2328257544133}a^{6}-\frac{28038620815650}{2328257544133}a^{5}-\frac{22617506148656}{2328257544133}a^{4}+\frac{69907726933390}{2328257544133}a^{3}+\frac{37427374092455}{2328257544133}a^{2}-\frac{29479559531629}{2328257544133}a-\frac{9738854803650}{2328257544133}$, $\frac{1049496746131}{2328257544133}a^{14}-\frac{4249575155725}{2328257544133}a^{13}+\frac{8000993702770}{2328257544133}a^{12}-\frac{19776577180236}{2328257544133}a^{11}+\frac{56975492704171}{2328257544133}a^{10}-\frac{100984549198382}{2328257544133}a^{9}+\frac{17616560734730}{2328257544133}a^{8}+\frac{148832776558315}{2328257544133}a^{7}-\frac{123755117658611}{2328257544133}a^{6}-\frac{8095199139875}{2328257544133}a^{5}-\frac{37055740357823}{2328257544133}a^{4}+\frac{49720924501894}{2328257544133}a^{3}+\frac{27438573459884}{2328257544133}a^{2}-\frac{17514774224470}{2328257544133}a-\frac{659456866622}{2328257544133}$, $\frac{292952766669}{2328257544133}a^{14}-\frac{834149796827}{2328257544133}a^{13}+\frac{1144564268545}{2328257544133}a^{12}-\frac{3951254014228}{2328257544133}a^{11}+\frac{11055772575791}{2328257544133}a^{10}-\frac{14236867825707}{2328257544133}a^{9}-\frac{14330025219917}{2328257544133}a^{8}+\frac{25327746559616}{2328257544133}a^{7}+\frac{6359205033623}{2328257544133}a^{6}-\frac{4692446767528}{2328257544133}a^{5}-\frac{26613128068049}{2328257544133}a^{4}-\frac{5299438870211}{2328257544133}a^{3}+\frac{10542446704233}{2328257544133}a^{2}+\frac{6876427063067}{2328257544133}a+\frac{1823742076844}{2328257544133}$
| 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: | \( 6179.9389969906615 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 6179.9389969906615 \cdot 1}{2\cdot\sqrt{24417546297445042591}}\cr\approx \mathstrut & 0.195953264325999 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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 - 4*x^14 + 7*x^13 - 17*x^12 + 51*x^11 - 87*x^10 - 7*x^9 + 172*x^8 - 100*x^7 - 69*x^6 - 17*x^5 + 62*x^4 + 50*x^3 - 22*x^2 - 18*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
$C_5\times S_3$ (as 15T4):
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 30 |
| The 15 conjugacy class representatives for $S_3 \times C_5$ |
| Character table for $S_3 \times C_5$ |
Intermediate fields
| 3.1.31.1, 5.5.923521.1 |
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
| Galois closure: | deg 30 |
| 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 | $15$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.3.0.1}{3} }^{5}$ | $15$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $15$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | R | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.5.4.1 | $x^{5} + 31$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 31.10.9.8 | $x^{10} + 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.31.10t1.a.a | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.31.5t1.a.a | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.31.5t1.a.b | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.31.10t1.a.b | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.31.10t1.a.c | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.31.5t1.a.c | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.31.5t1.a.d | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.31.10t1.a.d | $1$ | $ 31 $ | 10.0.26439622160671.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.31.3t2.b.a | $2$ | $ 31 $ | 3.1.31.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.961.15t4.a.a | $2$ | $ 31^{2}$ | 15.5.24417546297445042591.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.961.15t4.a.b | $2$ | $ 31^{2}$ | 15.5.24417546297445042591.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.961.15t4.a.c | $2$ | $ 31^{2}$ | 15.5.24417546297445042591.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.961.15t4.a.d | $2$ | $ 31^{2}$ | 15.5.24417546297445042591.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.