Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*x + 1)
gp: K = bnfinit(y^15 - 4*y^14 + 2*y^13 + 25*y^12 - 77*y^11 + 56*y^10 + 46*y^9 + 26*y^8 - 157*y^7 + 71*y^6 - 67*y^5 + 95*y^4 + 26*y^3 - 34*y^2 - 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*x + 1)
\( x^{15} - 4 x^{14} + 2 x^{13} + 25 x^{12} - 77 x^{11} + 56 x^{10} + 46 x^{9} + 26 x^{8} - 157 x^{7} + \cdots + 1 \)
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: |
\(-140994243189740741031\)
\(\medspace = -\,3^{5}\cdot 7^{5}\cdot 11^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.04\) | 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: | $3^{1/2}7^{1/2}11^{9/10}\approx 39.66094555677728$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-231}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4347605110091}a^{14}-\frac{818030110378}{4347605110091}a^{13}+\frac{602224352287}{4347605110091}a^{12}+\frac{1155803564900}{4347605110091}a^{11}+\frac{1326617349871}{4347605110091}a^{10}+\frac{1779848828027}{4347605110091}a^{9}-\frac{1382934097295}{4347605110091}a^{8}-\frac{754835061681}{4347605110091}a^{7}+\frac{1570309348036}{4347605110091}a^{6}-\frac{1812685449248}{4347605110091}a^{5}+\frac{2164895930079}{4347605110091}a^{4}+\frac{29412208359}{4347605110091}a^{3}-\frac{1328290112800}{4347605110091}a^{2}+\frac{878593468423}{4347605110091}a-\frac{1457639692692}{4347605110091}$
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: |
$\frac{505993223211}{4347605110091}a^{14}-\frac{2165786423665}{4347605110091}a^{13}+\frac{1269179994036}{4347605110091}a^{12}+\frac{13115276763520}{4347605110091}a^{11}-\frac{41901338888823}{4347605110091}a^{10}+\frac{32405293582463}{4347605110091}a^{9}+\frac{28213811855487}{4347605110091}a^{8}+\frac{10302413650486}{4347605110091}a^{7}-\frac{92203487904409}{4347605110091}a^{6}+\frac{34435996110183}{4347605110091}a^{5}-\frac{35365433717949}{4347605110091}a^{4}+\frac{57248865606028}{4347605110091}a^{3}+\frac{23949614566207}{4347605110091}a^{2}-\frac{16386561692700}{4347605110091}a-\frac{6245557483858}{4347605110091}$, $\frac{47982286398}{4347605110091}a^{14}-\frac{412182363761}{4347605110091}a^{13}+\frac{487166720943}{4347605110091}a^{12}+\frac{1966877008856}{4347605110091}a^{11}-\frac{8236438968421}{4347605110091}a^{10}+\frac{8513585070556}{4347605110091}a^{9}+\frac{10759296682309}{4347605110091}a^{8}-\frac{2116490020566}{4347605110091}a^{7}-\frac{32627881189617}{4347605110091}a^{6}+\frac{2015925252935}{4347605110091}a^{5}+\frac{4988778448683}{4347605110091}a^{4}+\frac{22771209483478}{4347605110091}a^{3}+\frac{3970765368354}{4347605110091}a^{2}-\frac{6229887604302}{4347605110091}a-\frac{7908670052214}{4347605110091}$, $\frac{627536500628}{4347605110091}a^{14}-\frac{2114563787176}{4347605110091}a^{13}+\frac{241452780021}{4347605110091}a^{12}+\frac{15095434812864}{4347605110091}a^{11}-\frac{39470121979315}{4347605110091}a^{10}+\frac{17249646933839}{4347605110091}a^{9}+\frac{26972504442543}{4347605110091}a^{8}+\frac{28935396507219}{4347605110091}a^{7}-\frac{71769324162169}{4347605110091}a^{6}+\frac{23205844644730}{4347605110091}a^{5}-\frac{31865071322439}{4347605110091}a^{4}+\frac{35689313971029}{4347605110091}a^{3}+\frac{11982340091341}{4347605110091}a^{2}-\frac{14847882754963}{4347605110091}a-\frac{1344517993204}{4347605110091}$, $\frac{805345238898}{4347605110091}a^{14}-\frac{2877055023800}{4347605110091}a^{13}+\frac{825582775164}{4347605110091}a^{12}+\frac{19363259312645}{4347605110091}a^{11}-\frac{54571962732572}{4347605110091}a^{10}+\frac{31891171533514}{4347605110091}a^{9}+\frac{31292402524894}{4347605110091}a^{8}+\frac{29133802245995}{4347605110091}a^{7}-\frac{96922242838230}{4347605110091}a^{6}+\frac{49515707406753}{4347605110091}a^{5}-\frac{51565081912673}{4347605110091}a^{4}+\frac{50971171538314}{4347605110091}a^{3}+\frac{13172310146841}{4347605110091}a^{2}-\frac{26242381223605}{4347605110091}a-\frac{1741680833857}{4347605110091}$, $\frac{159184183969}{4347605110091}a^{14}-\frac{1109356547731}{4347605110091}a^{13}+\frac{2446161172661}{4347605110091}a^{12}+\frac{2527336343398}{4347605110091}a^{11}-\frac{24371562185981}{4347605110091}a^{10}+\frac{49918922972229}{4347605110091}a^{9}-\frac{28457039820516}{4347605110091}a^{8}-\frac{17071807563031}{4347605110091}a^{7}-\frac{39583907062207}{4347605110091}a^{6}+\frac{97273932206205}{4347605110091}a^{5}-\frac{48482225244980}{4347605110091}a^{4}+\frac{66173116058494}{4347605110091}a^{3}-\frac{50732874851502}{4347605110091}a^{2}-\frac{14747542520238}{4347605110091}a+\frac{3823317567914}{4347605110091}$, $\frac{440648491328}{4347605110091}a^{14}-\frac{1109635202826}{4347605110091}a^{13}-\frac{711763106706}{4347605110091}a^{12}+\frac{9451234040088}{4347605110091}a^{11}-\frac{18752570832092}{4347605110091}a^{10}-\frac{2133943805938}{4347605110091}a^{9}+\frac{6126851295543}{4347605110091}a^{8}+\frac{43375593329864}{4347605110091}a^{7}-\frac{15216319796330}{4347605110091}a^{6}+\frac{3893283628350}{4347605110091}a^{5}-\frac{48645221326806}{4347605110091}a^{4}+\frac{1359753944396}{4347605110091}a^{3}-\frac{2710278271282}{4347605110091}a^{2}+\frac{14376624267948}{4347605110091}a+\frac{971589120295}{4347605110091}$, $\frac{397162840653}{4347605110091}a^{14}-\frac{1410842624342}{4347605110091}a^{13}+\frac{31834444682}{4347605110091}a^{12}+\frac{10513201011468}{4347605110091}a^{11}-\frac{26313714230500}{4347605110091}a^{10}+\frac{7139278323311}{4347605110091}a^{9}+\frac{32911015269713}{4347605110091}a^{8}+\frac{14646131939329}{4347605110091}a^{7}-\frac{62156160243745}{4347605110091}a^{6}+\frac{3045643010302}{4347605110091}a^{5}-\frac{300047561728}{4347605110091}a^{4}+\frac{18030459271801}{4347605110091}a^{3}+\frac{25608091424263}{4347605110091}a^{2}-\frac{12313566526702}{4347605110091}a-\frac{15763289715825}{4347605110091}$, $\frac{2009963876369}{4347605110091}a^{14}-\frac{6077873915505}{4347605110091}a^{13}-\frac{1645763799006}{4347605110091}a^{12}+\frac{48116166389276}{4347605110091}a^{11}-\frac{108617929099993}{4347605110091}a^{10}+\frac{12176486945092}{4347605110091}a^{9}+\frac{96121240187510}{4347605110091}a^{8}+\frac{137621004290795}{4347605110091}a^{7}-\frac{175826834278572}{4347605110091}a^{6}-\frac{491517866756}{4347605110091}a^{5}-\frac{137175022836414}{4347605110091}a^{4}+\frac{48782766969958}{4347605110091}a^{3}+\frac{78270618272074}{4347605110091}a^{2}+\frac{5961081107174}{4347605110091}a-\frac{5571028818492}{4347605110091}$, $\frac{2382259470}{48849495619}a^{14}-\frac{19495409769}{48849495619}a^{13}+\frac{36498508732}{48849495619}a^{12}+\frac{62931648228}{48849495619}a^{11}-\frac{421091174963}{48849495619}a^{10}+\frac{705749210786}{48849495619}a^{9}-\frac{42032029050}{48849495619}a^{8}-\frac{358906014522}{48849495619}a^{7}-\frac{1057036660123}{48849495619}a^{6}+\frac{1100298046326}{48849495619}a^{5}-\frac{261187041008}{48849495619}a^{4}+\frac{1033373498818}{48849495619}a^{3}-\frac{295102700294}{48849495619}a^{2}-\frac{384025009449}{48849495619}a+\frac{21440319340}{48849495619}$
| 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: | \( 18274.701264234773 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 18274.701264234773 \cdot 1}{2\cdot\sqrt{140994243189740741031}}\cr\approx \mathstrut & 0.241139813327711 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*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 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*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 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*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 + 2*x^13 + 25*x^12 - 77*x^11 + 56*x^10 + 46*x^9 + 26*x^8 - 157*x^7 + 71*x^6 - 67*x^5 + 95*x^4 + 26*x^3 - 34*x^2 - 11*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
| 3.1.231.1, \(\Q(\zeta_{11})^+\) |
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: | 30.0.893083200934004322334040290940494739136293593671.1 |
| 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 | $15$ | R | $15$ | R | R | $15$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $15$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | $15$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $15$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(7\)
| 7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.231.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | \(\Q(\sqrt{-231}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.231.10t1.b.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | 10.0.9630096522760791.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.231.10t1.b.b | $1$ | $ 3 \cdot 7 \cdot 11 $ | 10.0.9630096522760791.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.231.10t1.b.c | $1$ | $ 3 \cdot 7 \cdot 11 $ | 10.0.9630096522760791.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.231.10t1.b.d | $1$ | $ 3 \cdot 7 \cdot 11 $ | 10.0.9630096522760791.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.231.3t2.a.a | $2$ | $ 3 \cdot 7 \cdot 11 $ | 3.1.231.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2541.15t4.b.a | $2$ | $ 3 \cdot 7 \cdot 11^{2}$ | 15.5.140994243189740741031.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.2541.15t4.b.b | $2$ | $ 3 \cdot 7 \cdot 11^{2}$ | 15.5.140994243189740741031.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.2541.15t4.b.c | $2$ | $ 3 \cdot 7 \cdot 11^{2}$ | 15.5.140994243189740741031.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ |
| * | 2.2541.15t4.b.d | $2$ | $ 3 \cdot 7 \cdot 11^{2}$ | 15.5.140994243189740741031.1 | $S_3 \times C_5$ (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 *.