Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27)
gp: K = bnfinit(y^15 - 4*y^14 - 5*y^13 + 43*y^12 - 64*y^11 - 63*y^10 + 340*y^9 - 375*y^8 - 161*y^7 + 842*y^6 - 821*y^5 + 86*y^4 + 384*y^3 - 294*y^2 + 117*y - 27, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27)
\( x^{15} - 4 x^{14} - 5 x^{13} + 43 x^{12} - 64 x^{11} - 63 x^{10} + 340 x^{9} - 375 x^{8} - 161 x^{7} + \cdots - 27 \)
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: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-12274698536106109012992\)
\(\medspace = -\,2^{10}\cdot 3\cdot 31^{2}\cdot 401^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.69\) | 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: | $2^{2/3}3^{1/2}31^{2/3}401^{1/2}\approx 543.3261484593049$ | ||
| Ramified primes: |
\(2\), \(3\), \(31\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{132930969579}a^{14}+\frac{20895671954}{132930969579}a^{13}+\frac{53486872747}{132930969579}a^{12}+\frac{31628353234}{132930969579}a^{11}-\frac{54448864366}{132930969579}a^{10}+\frac{15668018797}{44310323193}a^{9}+\frac{24143923969}{132930969579}a^{8}+\frac{2751766680}{14770107731}a^{7}+\frac{57376801486}{132930969579}a^{6}-\frac{25283214547}{132930969579}a^{5}-\frac{26964436553}{132930969579}a^{4}-\frac{26870251579}{132930969579}a^{3}+\frac{169767914}{14770107731}a^{2}-\frac{5472816758}{44310323193}a-\frac{4970091507}{14770107731}$
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: | $11$ | 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: |
$\frac{350255762}{5779607373}a^{14}-\frac{611088224}{5779607373}a^{13}-\frac{3711989893}{5779607373}a^{12}+\frac{7874268770}{5779607373}a^{11}+\frac{1251219610}{5779607373}a^{10}-\frac{3815236366}{642178597}a^{9}+\frac{43821736490}{5779607373}a^{8}+\frac{8706478796}{1926535791}a^{7}-\frac{93857177368}{5779607373}a^{6}+\frac{70908419794}{5779607373}a^{5}+\frac{46795920734}{5779607373}a^{4}-\frac{62657158028}{5779607373}a^{3}-\frac{3103586306}{1926535791}a^{2}-\frac{171928204}{1926535791}a-\frac{297778548}{642178597}$, $\frac{350255762}{5779607373}a^{14}-\frac{611088224}{5779607373}a^{13}-\frac{3711989893}{5779607373}a^{12}+\frac{7874268770}{5779607373}a^{11}+\frac{1251219610}{5779607373}a^{10}-\frac{3815236366}{642178597}a^{9}+\frac{43821736490}{5779607373}a^{8}+\frac{8706478796}{1926535791}a^{7}-\frac{93857177368}{5779607373}a^{6}+\frac{70908419794}{5779607373}a^{5}+\frac{46795920734}{5779607373}a^{4}-\frac{62657158028}{5779607373}a^{3}-\frac{3103586306}{1926535791}a^{2}-\frac{171928204}{1926535791}a+\frac{344400049}{642178597}$, $\frac{87294815}{132930969579}a^{14}+\frac{1585236508}{132930969579}a^{13}-\frac{1327321429}{132930969579}a^{12}-\frac{15038710717}{132930969579}a^{11}+\frac{1120216351}{132930969579}a^{10}-\frac{2141058256}{44310323193}a^{9}-\frac{22624804483}{132930969579}a^{8}+\frac{1631057641}{14770107731}a^{7}-\frac{91242023275}{132930969579}a^{6}+\frac{161242123948}{132930969579}a^{5}+\frac{102494241692}{132930969579}a^{4}-\frac{216886018385}{132930969579}a^{3}+\frac{19961835633}{14770107731}a^{2}+\frac{16699309877}{44310323193}a-\frac{3631964764}{14770107731}$, $\frac{57341852}{642178597}a^{14}-\frac{369768650}{1926535791}a^{13}-\frac{1621331524}{1926535791}a^{12}+\frac{4480320490}{1926535791}a^{11}-\frac{1929031766}{1926535791}a^{10}-\frac{15320674180}{1926535791}a^{9}+\frac{10031076227}{642178597}a^{8}-\frac{5238488306}{1926535791}a^{7}-\frac{14738600454}{642178597}a^{6}+\frac{66553468054}{1926535791}a^{5}-\frac{15943886959}{1926535791}a^{4}-\frac{33277540070}{1926535791}a^{3}+\frac{23964491624}{1926535791}a^{2}-\frac{3583423620}{642178597}a+\frac{885839333}{642178597}$, $\frac{91592112}{14770107731}a^{14}+\frac{1945729190}{44310323193}a^{13}-\frac{10591039472}{44310323193}a^{12}-\frac{13687153630}{44310323193}a^{11}+\frac{90922828286}{44310323193}a^{10}-\frac{93174307700}{44310323193}a^{9}-\frac{66248543360}{14770107731}a^{8}+\frac{645250581563}{44310323193}a^{7}-\frac{126546997170}{14770107731}a^{6}-\frac{722751795679}{44310323193}a^{5}+\frac{1462101557959}{44310323193}a^{4}-\frac{720555868159}{44310323193}a^{3}-\frac{673957896512}{44310323193}a^{2}+\frac{228599633650}{14770107731}a-\frac{43146156764}{14770107731}$, $a-1$, $\frac{7672064374}{132930969579}a^{14}-\frac{18565602172}{132930969579}a^{13}-\frac{79715720945}{132930969579}a^{12}+\frac{230917614709}{132930969579}a^{11}+\frac{9873624650}{132930969579}a^{10}-\frac{272129854460}{44310323193}a^{9}+\frac{1168177775137}{132930969579}a^{8}+\frac{38507497377}{14770107731}a^{7}-\frac{2150662366448}{132930969579}a^{6}+\frac{1772667845732}{132930969579}a^{5}-\frac{25827452558}{132930969579}a^{4}-\frac{849600669643}{132930969579}a^{3}+\frac{66930328369}{14770107731}a^{2}-\frac{86128169648}{44310323193}a+\frac{251403995}{14770107731}$, $\frac{4474514918}{132930969579}a^{14}-\frac{18094948097}{132930969579}a^{13}-\frac{25573928026}{132930969579}a^{12}+\frac{200768857955}{132930969579}a^{11}-\frac{255767740055}{132930969579}a^{10}-\frac{124121349814}{44310323193}a^{9}+\frac{1559624632853}{132930969579}a^{8}-\frac{138707025053}{14770107731}a^{7}-\frac{1261917479662}{132930969579}a^{6}+\frac{3481420770283}{132930969579}a^{5}-\frac{2282840640034}{132930969579}a^{4}-\frac{599827653707}{132930969579}a^{3}+\frac{74738281823}{14770107731}a^{2}-\frac{150450822436}{44310323193}a+\frac{13044781505}{14770107731}$, $\frac{19256018816}{132930969579}a^{14}-\frac{54949655618}{132930969579}a^{13}-\frac{155266600744}{132930969579}a^{12}+\frac{640190655545}{132930969579}a^{11}-\frac{547708227545}{132930969579}a^{10}-\frac{189390480751}{14770107731}a^{9}+\frac{4679182864328}{132930969579}a^{8}-\frac{814249843225}{44310323193}a^{7}-\frac{5364849933187}{132930969579}a^{6}+\frac{10729829825146}{132930969579}a^{5}-\frac{5120813117866}{132930969579}a^{4}-\frac{3670158072347}{132930969579}a^{3}+\frac{1218465323407}{44310323193}a^{2}-\frac{584177236492}{44310323193}a+\frac{60099873361}{14770107731}$, $\frac{26574738068}{132930969579}a^{14}-\frac{49494725810}{132930969579}a^{13}-\frac{269140279087}{132930969579}a^{12}+\frac{626554206950}{132930969579}a^{11}-\frac{49665303983}{132930969579}a^{10}-\frac{282001067123}{14770107731}a^{9}+\frac{3646807434941}{132930969579}a^{8}+\frac{298310969684}{44310323193}a^{7}-\frac{6747688520935}{132930969579}a^{6}+\frac{6797366270317}{132930969579}a^{5}+\frac{1086019031390}{132930969579}a^{4}-\frac{3605508539198}{132930969579}a^{3}+\frac{448538981452}{44310323193}a^{2}-\frac{152295823033}{44310323193}a+\frac{8897031453}{14770107731}$, $\frac{13676620850}{132930969579}a^{14}-\frac{22969102541}{132930969579}a^{13}-\frac{149187675199}{132930969579}a^{12}+\frac{297758873387}{132930969579}a^{11}+\frac{108317556451}{132930969579}a^{10}-\frac{146616680519}{14770107731}a^{9}+\frac{1461548095814}{132930969579}a^{8}+\frac{356902521047}{44310323193}a^{7}-\frac{3248058762586}{132930969579}a^{6}+\frac{2202231110926}{132930969579}a^{5}+\frac{1142325102953}{132930969579}a^{4}-\frac{1449575925827}{132930969579}a^{3}+\frac{69199942033}{44310323193}a^{2}-\frac{96665800081}{44310323193}a-\frac{4658702676}{14770107731}$
| 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: | \( 791207.326098 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 791207.326098 \cdot 1}{2\cdot\sqrt{12274698536106109012992}}\cr\approx \mathstrut & 0.453486593977 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27)
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 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27, 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 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27);
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 - 5*x^13 + 43*x^12 - 64*x^11 - 63*x^10 + 340*x^9 - 375*x^8 - 161*x^7 + 842*x^6 - 821*x^5 + 86*x^4 + 384*x^3 - 294*x^2 + 117*x - 27);
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
$S_3\wr D_5$ (as 15T86):
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 77760 |
| The 72 conjugacy class representatives for $S_3\wr D_5$ |
| Character table for $S_3\wr D_5$ |
Intermediate fields
| 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 |
| Degree 45 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 | R | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | $15$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.15.10.1 | $x^{15} + 13 x^{12} + 3 x^{10} + 13 x^{9} - 174 x^{7} - 45 x^{6} + 3 x^{5} + 513 x^{4} + 63 x^{2} - 216 x + 109$ | $3$ | $5$ | $10$ | $S_3 \times C_5$ | $[\ ]_{3}^{10}$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.3.2.1 | $x^{3} + 31$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |