Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3)
gp: K = bnfinit(y^15 - 3*y^14 - y^13 - 4*y^12 + 10*y^11 + 25*y^10 + 36*y^9 - 46*y^8 - 111*y^7 - 151*y^6 + 138*y^5 + 181*y^4 - 15*y^3 - 52*y^2 - 24*y + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3)
\( x^{15} - 3 x^{14} - x^{13} - 4 x^{12} + 10 x^{11} + 25 x^{10} + 36 x^{9} - 46 x^{8} - 111 x^{7} + \cdots + 3 \)
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: |
\(-405150968993137160643\)
\(\medspace = -\,3^{5}\cdot 401^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.65\) | 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: | $3^{1/2}401^{1/2}\approx 34.68429039204925$ | ||
| Ramified primes: |
\(3\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1203}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{2}{9}a^{5}+\frac{1}{9}a^{4}+\frac{4}{9}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{12}+\frac{4}{27}a^{11}-\frac{1}{9}a^{10}-\frac{4}{27}a^{9}-\frac{4}{27}a^{8}+\frac{7}{27}a^{6}-\frac{7}{27}a^{5}+\frac{10}{27}a^{4}+\frac{10}{27}a^{3}+\frac{10}{27}a^{2}+\frac{1}{9}a-\frac{4}{9}$, $\frac{1}{98193605913}a^{14}-\frac{48306514}{98193605913}a^{13}-\frac{979406726}{98193605913}a^{12}-\frac{562712635}{10910400657}a^{11}-\frac{765327298}{98193605913}a^{10}-\frac{15682032163}{98193605913}a^{9}+\frac{8208629555}{32731201971}a^{8}-\frac{833699642}{7553354301}a^{7}-\frac{27097251994}{98193605913}a^{6}+\frac{37438457428}{98193605913}a^{5}+\frac{38930006218}{98193605913}a^{4}-\frac{7805586485}{98193605913}a^{3}+\frac{1009784150}{32731201971}a^{2}-\frac{7357296181}{32731201971}a-\frac{4734927202}{10910400657}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{9094778288}{98193605913}a^{14}-\frac{5921157496}{32731201971}a^{13}-\frac{9774986551}{32731201971}a^{12}-\frac{63787393663}{98193605913}a^{11}+\frac{29120607376}{98193605913}a^{10}+\frac{269804071703}{98193605913}a^{9}+\frac{602166941569}{98193605913}a^{8}+\frac{12967157345}{7553354301}a^{7}-\frac{317739319174}{32731201971}a^{6}-\frac{791940522556}{32731201971}a^{5}-\frac{1072094870546}{98193605913}a^{4}+\frac{958320855859}{98193605913}a^{3}+\frac{1050413684696}{98193605913}a^{2}+\frac{47017533178}{10910400657}a-\frac{9581417387}{32731201971}$, $\frac{559971073}{98193605913}a^{14}-\frac{3208563947}{98193605913}a^{13}-\frac{1040637157}{98193605913}a^{12}+\frac{11566225538}{98193605913}a^{11}+\frac{22352947352}{98193605913}a^{10}+\frac{1064718127}{3636800219}a^{9}-\frac{48482738201}{98193605913}a^{8}-\frac{16486607264}{7553354301}a^{7}-\frac{265322573717}{98193605913}a^{6}+\frac{129899763401}{98193605913}a^{5}+\frac{271910339606}{32731201971}a^{4}+\frac{107091013391}{10910400657}a^{3}-\frac{243244734805}{98193605913}a^{2}-\frac{208827816833}{32731201971}a-\frac{55775876141}{32731201971}$, $\frac{9094778288}{98193605913}a^{14}-\frac{5921157496}{32731201971}a^{13}-\frac{9774986551}{32731201971}a^{12}-\frac{63787393663}{98193605913}a^{11}+\frac{29120607376}{98193605913}a^{10}+\frac{269804071703}{98193605913}a^{9}+\frac{602166941569}{98193605913}a^{8}+\frac{12967157345}{7553354301}a^{7}-\frac{317739319174}{32731201971}a^{6}-\frac{791940522556}{32731201971}a^{5}-\frac{1072094870546}{98193605913}a^{4}+\frac{958320855859}{98193605913}a^{3}+\frac{1050413684696}{98193605913}a^{2}+\frac{47017533178}{10910400657}a-\frac{42312619358}{32731201971}$, $\frac{97902853}{7553354301}a^{14}-\frac{458355365}{7553354301}a^{13}+\frac{313325546}{7553354301}a^{12}-\frac{80205466}{7553354301}a^{11}+\frac{1953830279}{7553354301}a^{10}+\frac{427029340}{2517784767}a^{9}-\frac{683210141}{7553354301}a^{8}-\frac{12580385360}{7553354301}a^{7}-\frac{8505658838}{7553354301}a^{6}+\frac{2052349757}{7553354301}a^{5}+\frac{5215434892}{839261589}a^{4}+\frac{4157455448}{2517784767}a^{3}-\frac{25792444885}{7553354301}a^{2}-\frac{4866104453}{2517784767}a-\frac{1219306508}{2517784767}$, $\frac{131647510}{10910400657}a^{14}+\frac{5700236573}{98193605913}a^{13}-\frac{28082778674}{98193605913}a^{12}-\frac{10597850698}{98193605913}a^{11}-\frac{14268743540}{32731201971}a^{10}+\frac{112309172947}{98193605913}a^{9}+\frac{250027566826}{98193605913}a^{8}+\frac{993244261}{279753863}a^{7}-\frac{389005393243}{98193605913}a^{6}-\frac{976784935454}{98193605913}a^{5}-\frac{1476052388614}{98193605913}a^{4}+\frac{836768981558}{98193605913}a^{3}+\frac{694274481458}{98193605913}a^{2}+\frac{100704353414}{32731201971}a-\frac{25081561202}{32731201971}$, $\frac{8083196705}{98193605913}a^{14}-\frac{16745541115}{98193605913}a^{13}-\frac{23856176267}{98193605913}a^{12}-\frac{52906892405}{98193605913}a^{11}+\frac{28538904337}{98193605913}a^{10}+\frac{25759577365}{10910400657}a^{9}+\frac{498136700177}{98193605913}a^{8}+\frac{7474772726}{7553354301}a^{7}-\frac{806197303918}{98193605913}a^{6}-\frac{1944375642074}{98193605913}a^{5}-\frac{26624122602}{3636800219}a^{4}+\frac{29221666235}{3636800219}a^{3}+\frac{521815296541}{98193605913}a^{2}+\frac{60458953985}{32731201971}a-\frac{3795987466}{32731201971}$, $\frac{238476115}{98193605913}a^{14}-\frac{3501269351}{32731201971}a^{13}+\frac{951000824}{3636800219}a^{12}+\frac{15011996332}{98193605913}a^{11}+\frac{54565598603}{98193605913}a^{10}-\frac{68495469545}{98193605913}a^{9}-\frac{243260349910}{98193605913}a^{8}-\frac{36251170925}{7553354301}a^{7}+\frac{61499524126}{32731201971}a^{6}+\frac{107237358461}{10910400657}a^{5}+\frac{1902237847178}{98193605913}a^{4}-\frac{361492803598}{98193605913}a^{3}-\frac{1285029361190}{98193605913}a^{2}-\frac{12480752500}{3636800219}a+\frac{10776624806}{32731201971}$, $\frac{9444556570}{98193605913}a^{14}-\frac{20695927874}{98193605913}a^{13}-\frac{30242569636}{98193605913}a^{12}-\frac{46025451529}{98193605913}a^{11}+\frac{46399254404}{98193605913}a^{10}+\frac{98504322056}{32731201971}a^{9}+\frac{521755662487}{98193605913}a^{8}-\frac{5038959242}{7553354301}a^{7}-\frac{1175696117333}{98193605913}a^{6}-\frac{2107658068018}{98193605913}a^{5}-\frac{69643683965}{32731201971}a^{4}+\frac{627792010771}{32731201971}a^{3}+\frac{421489086275}{98193605913}a^{2}+\frac{3992610160}{32731201971}a-\frac{25403056160}{32731201971}$, $\frac{185975050}{32731201971}a^{14}-\frac{205163650}{10910400657}a^{13}+\frac{2554711}{3636800219}a^{12}-\frac{895427270}{32731201971}a^{11}+\frac{2004587798}{32731201971}a^{10}+\frac{4909989742}{32731201971}a^{9}+\frac{6216011738}{32731201971}a^{8}-\frac{662189027}{2517784767}a^{7}-\frac{1762727650}{3636800219}a^{6}-\frac{4169491289}{3636800219}a^{5}+\frac{21850012352}{32731201971}a^{4}+\frac{3360386099}{32731201971}a^{3}+\frac{10561846471}{32731201971}a^{2}+\frac{1036032300}{3636800219}a-\frac{3831541105}{10910400657}$
| 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: | \( 77823.7688685 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 77823.7688685 \cdot 1}{2\cdot\sqrt{405150968993137160643}}\cr\approx \mathstrut & 0.605791454275 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3)
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 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3, 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 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3);
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 - 3*x^14 - x^13 - 4*x^12 + 10*x^11 + 25*x^10 + 36*x^9 - 46*x^8 - 111*x^7 - 151*x^6 + 138*x^5 + 181*x^4 - 15*x^3 - 52*x^2 - 24*x + 3);
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
| 3.1.1203.1, 5.5.160801.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $15$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |