Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*x - 1)
gp: K = bnfinit(y^15 - 2*y^14 + y^12 - 6*y^11 + 27*y^9 - 37*y^8 + 44*y^7 - 34*y^6 + 14*y^5 - 15*y^4 - 8*y^3 + 9*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*x - 1)
\( x^{15} - 2 x^{14} + x^{12} - 6 x^{11} + 27 x^{9} - 37 x^{8} + 44 x^{7} - 34 x^{6} + 14 x^{5} - 15 x^{4} + \cdots - 1 \)
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: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(351730374981194881\)
\(\medspace = 7^{12}\cdot 71^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.78\) | 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: | $7^{5/6}71^{4/5}\approx 153.19868787917656$ | ||
| Ramified primes: |
\(7\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{7}+\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{2049272407}a^{14}+\frac{650660}{2049272407}a^{13}+\frac{38608274}{2049272407}a^{12}-\frac{215400421}{2049272407}a^{11}+\frac{13067319}{292753201}a^{10}+\frac{715149348}{2049272407}a^{9}-\frac{754312263}{2049272407}a^{8}-\frac{413178839}{2049272407}a^{7}-\frac{648259}{1899233}a^{6}+\frac{481581404}{2049272407}a^{5}-\frac{790941285}{2049272407}a^{4}+\frac{390461632}{2049272407}a^{3}-\frac{293010650}{2049272407}a^{2}-\frac{642956668}{2049272407}a+\frac{365753205}{2049272407}$
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: | $8$ | 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: |
$\frac{255880921}{2049272407}a^{14}-\frac{485636051}{2049272407}a^{13}-\frac{197206}{2049272407}a^{12}+\frac{167332587}{2049272407}a^{11}-\frac{1544287239}{2049272407}a^{10}-\frac{116046345}{2049272407}a^{9}+\frac{6574703196}{2049272407}a^{8}-\frac{8781867005}{2049272407}a^{7}+\frac{11059264}{1899233}a^{6}-\frac{1313768850}{292753201}a^{5}+\frac{4266157921}{2049272407}a^{4}-\frac{4294261913}{2049272407}a^{3}-\frac{3596720318}{2049272407}a^{2}-\frac{346727385}{2049272407}a+\frac{540418521}{292753201}$, $\frac{255880921}{2049272407}a^{14}-\frac{485636051}{2049272407}a^{13}-\frac{197206}{2049272407}a^{12}+\frac{167332587}{2049272407}a^{11}-\frac{1544287239}{2049272407}a^{10}-\frac{116046345}{2049272407}a^{9}+\frac{6574703196}{2049272407}a^{8}-\frac{8781867005}{2049272407}a^{7}+\frac{11059264}{1899233}a^{6}-\frac{1313768850}{292753201}a^{5}+\frac{4266157921}{2049272407}a^{4}-\frac{4294261913}{2049272407}a^{3}-\frac{3596720318}{2049272407}a^{2}-\frac{346727385}{2049272407}a+\frac{247665320}{292753201}$, $\frac{12852236}{292753201}a^{14}-\frac{122422434}{2049272407}a^{13}-\frac{114538017}{2049272407}a^{12}+\frac{151620566}{2049272407}a^{11}-\frac{529271406}{2049272407}a^{10}-\frac{369366035}{2049272407}a^{9}+\frac{316189996}{292753201}a^{8}-\frac{2148532518}{2049272407}a^{7}+\frac{1578590}{1899233}a^{6}+\frac{439905178}{2049272407}a^{5}-\frac{1281000761}{2049272407}a^{4}+\frac{3334239357}{2049272407}a^{3}-\frac{684134664}{292753201}a^{2}+\frac{977574347}{2049272407}a-\frac{129883449}{292753201}$, $\frac{230234853}{2049272407}a^{14}-\frac{423783931}{2049272407}a^{13}+\frac{5909252}{292753201}a^{12}+\frac{21717355}{292753201}a^{11}-\frac{1643983526}{2049272407}a^{10}-\frac{154507558}{2049272407}a^{9}+\frac{823087748}{292753201}a^{8}-\frac{8487705271}{2049272407}a^{7}+\frac{10976977}{1899233}a^{6}-\frac{6001607020}{2049272407}a^{5}+\frac{1515621637}{2049272407}a^{4}-\frac{1306943656}{2049272407}a^{3}-\frac{2919595537}{2049272407}a^{2}-\frac{2254342921}{2049272407}a-\frac{513987218}{2049272407}$, $\frac{62648636}{2049272407}a^{14}+\frac{5792520}{2049272407}a^{13}-\frac{224814635}{2049272407}a^{12}-\frac{41965270}{2049272407}a^{11}-\frac{198590085}{2049272407}a^{10}-\frac{785041694}{2049272407}a^{9}+\frac{1556176147}{2049272407}a^{8}+\frac{1553912298}{2049272407}a^{7}-\frac{1058661}{1899233}a^{6}+\frac{1558469032}{2049272407}a^{5}-\frac{1083243712}{2049272407}a^{4}-\frac{3513994917}{2049272407}a^{3}-\frac{762093273}{2049272407}a^{2}-\frac{49549904}{292753201}a+\frac{572514282}{2049272407}$, $\frac{221055444}{2049272407}a^{14}-\frac{347237069}{2049272407}a^{13}-\frac{219151923}{2049272407}a^{12}+\frac{328922297}{2049272407}a^{11}-\frac{1314313100}{2049272407}a^{10}-\frac{504703060}{2049272407}a^{9}+\frac{6085241802}{2049272407}a^{8}-\frac{6047367721}{2049272407}a^{7}+\frac{4997054}{1899233}a^{6}-\frac{1520419438}{2049272407}a^{5}-\frac{3855266138}{2049272407}a^{4}+\frac{3073335172}{2049272407}a^{3}-\frac{1026437769}{292753201}a^{2}+\frac{986250905}{2049272407}a+\frac{1202736900}{2049272407}$, $\frac{33247931}{292753201}a^{14}-\frac{464036849}{2049272407}a^{13}+\frac{23194359}{292753201}a^{12}+\frac{998838}{292753201}a^{11}-\frac{205663050}{292753201}a^{10}+\frac{155140392}{2049272407}a^{9}+\frac{5132102050}{2049272407}a^{8}-\frac{9330341444}{2049272407}a^{7}+\frac{1825530}{271319}a^{6}-\frac{13121056801}{2049272407}a^{5}+\frac{8242550949}{2049272407}a^{4}-\frac{5076936499}{2049272407}a^{3}-\frac{1777739870}{2049272407}a^{2}-\frac{653214006}{2049272407}a+\frac{1124025486}{2049272407}$, $\frac{2518059}{157636339}a^{14}-\frac{4280195}{157636339}a^{13}+\frac{962023}{157636339}a^{12}+\frac{2944593}{157636339}a^{11}-\frac{14921471}{157636339}a^{10}-\frac{12044792}{157636339}a^{9}+\frac{50284490}{157636339}a^{8}-\frac{96676948}{157636339}a^{7}+\frac{1052425}{1899233}a^{6}-\frac{72658038}{157636339}a^{5}+\frac{92140511}{157636339}a^{4}-\frac{141206899}{157636339}a^{3}+\frac{86490816}{157636339}a^{2}+\frac{24219284}{157636339}a+\frac{90969295}{157636339}$
| 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: | \( 694.485336338 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 694.485336338 \cdot 1}{2\cdot\sqrt{351730374981194881}}\cr\approx \mathstrut & 0.288202234874 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*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 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*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 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*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 - 2*x^14 + x^12 - 6*x^11 + 27*x^9 - 37*x^8 + 44*x^7 - 34*x^6 + 14*x^5 - 15*x^4 - 8*x^3 + 9*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
$C_5^3:C_6$ (as 15T30):
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 750 |
| The 32 conjugacy class representatives for $C_5^3:C_6$ |
| Character table for $C_5^3:C_6$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
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 15 siblings: | data not computed |
| 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 | $15$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $15$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $15$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.5.0.1}{5} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $15$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(71\)
| 71.5.4.2 | $x^{5} + 142$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |