Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4)
gp: K = bnfinit(y^15 - 3*y^13 - 2*y^12 + 12*y^10 + 50*y^9 - 54*y^7 + 68*y^6 - 162*y^5 + 30*y^4 - 67*y^3 + 15*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4)
\( x^{15} - 3x^{13} - 2x^{12} + 12x^{10} + 50x^{9} - 54x^{7} + 68x^{6} - 162x^{5} + 30x^{4} - 67x^{3} + 15x + 4 \)
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: |
\(30638157055420022784\)
\(\medspace = 2^{14}\cdot 3^{18}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.91\) | 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: | $2^{7/6}3^{25/18}13^{1/2}\approx 37.22560348698383$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{2}{9}a^{5}-\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{18}a^{12}-\frac{1}{18}a^{11}-\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{18}a^{6}-\frac{7}{18}a^{5}+\frac{5}{18}a^{4}-\frac{7}{18}a^{3}+\frac{5}{18}a^{2}-\frac{7}{18}a-\frac{1}{9}$, $\frac{1}{702}a^{13}-\frac{8}{351}a^{12}-\frac{1}{117}a^{11}-\frac{2}{27}a^{10}+\frac{23}{351}a^{9}+\frac{19}{39}a^{8}+\frac{146}{351}a^{7}+\frac{58}{351}a^{6}-\frac{53}{117}a^{5}+\frac{4}{39}a^{4}+\frac{14}{117}a^{3}-\frac{17}{39}a^{2}-\frac{79}{702}a+\frac{128}{351}$, $\frac{1}{5980356252}a^{14}-\frac{646705}{5980356252}a^{13}+\frac{21766499}{996726042}a^{12}+\frac{37570363}{2990178126}a^{11}+\frac{245957080}{1495089063}a^{10}-\frac{78032023}{996726042}a^{9}+\frac{394532939}{2990178126}a^{8}-\frac{195086908}{1495089063}a^{7}+\frac{7957133}{498363021}a^{6}+\frac{103565231}{996726042}a^{5}-\frac{290159579}{996726042}a^{4}-\frac{10850479}{996726042}a^{3}-\frac{1168436515}{5980356252}a^{2}+\frac{98787985}{460027404}a+\frac{238537894}{498363021}$
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: |
$\frac{7448651}{332242014}a^{14}+\frac{27258287}{996726042}a^{13}-\frac{9734225}{110747338}a^{12}-\frac{38053943}{332242014}a^{11}+\frac{28343929}{996726042}a^{10}+\frac{88592663}{332242014}a^{9}+\frac{451740131}{332242014}a^{8}+\frac{1088636987}{996726042}a^{7}-\frac{228260011}{110747338}a^{6}+\frac{311036189}{332242014}a^{5}+\frac{24404105}{110747338}a^{4}-\frac{2017122415}{332242014}a^{3}+\frac{365268223}{166121007}a^{2}-\frac{1332344296}{498363021}a-\frac{110744081}{166121007}$, $\frac{434437579}{5980356252}a^{14}-\frac{171030521}{5980356252}a^{13}-\frac{318289718}{1495089063}a^{12}-\frac{7331551}{115006851}a^{11}+\frac{121982017}{2990178126}a^{10}+\frac{1299379787}{1495089063}a^{9}+\frac{4950313660}{1495089063}a^{8}-\frac{4102571839}{2990178126}a^{7}-\frac{11117785951}{2990178126}a^{6}+\frac{1038475940}{166121007}a^{5}-\frac{781246083}{55373669}a^{4}+\frac{3699677230}{498363021}a^{3}-\frac{39500013583}{5980356252}a^{2}+\frac{11724796379}{5980356252}a+\frac{131341058}{115006851}$, $\frac{351331147}{5980356252}a^{14}+\frac{104314511}{5980356252}a^{13}-\frac{27306053}{166121007}a^{12}-\frac{252048856}{1495089063}a^{11}-\frac{189816205}{2990178126}a^{10}+\frac{342903532}{498363021}a^{9}+\frac{4695165091}{1495089063}a^{8}+\frac{2903261701}{2990178126}a^{7}-\frac{2594329315}{996726042}a^{6}+\frac{175322923}{55373669}a^{5}-\frac{4261104197}{498363021}a^{4}+\frac{8301097}{166121007}a^{3}-\frac{30112846747}{5980356252}a^{2}-\frac{4167882473}{5980356252}a+\frac{228229804}{498363021}$, $\frac{12525497}{664484028}a^{14}+\frac{273835877}{5980356252}a^{13}-\frac{111579902}{1495089063}a^{12}-\frac{7736561}{38335617}a^{11}-\frac{55346275}{2990178126}a^{10}+\frac{548773211}{1495089063}a^{9}+\frac{245484410}{166121007}a^{8}+\frac{5897082367}{2990178126}a^{7}-\frac{6692528359}{2990178126}a^{6}-\frac{126593446}{55373669}a^{5}+\frac{1103037968}{498363021}a^{4}-\frac{2782687304}{498363021}a^{3}-\frac{230662423}{1993452084}a^{2}+\frac{7940692933}{5980356252}a+\frac{58285676}{115006851}$, $\frac{85811095}{5980356252}a^{14}+\frac{38672809}{5980356252}a^{13}-\frac{78762698}{1495089063}a^{12}-\frac{54826717}{1495089063}a^{11}+\frac{25068337}{2990178126}a^{10}+\frac{237438353}{1495089063}a^{9}+\frac{1192436668}{1495089063}a^{8}+\frac{670962101}{2990178126}a^{7}-\frac{3429415903}{2990178126}a^{6}+\frac{541790000}{498363021}a^{5}-\frac{838310248}{498363021}a^{4}-\frac{278490376}{166121007}a^{3}+\frac{14499049469}{5980356252}a^{2}-\frac{14877943219}{5980356252}a+\frac{2891670290}{1495089063}$, $\frac{35149801}{1993452084}a^{14}+\frac{95848729}{5980356252}a^{13}-\frac{194727023}{2990178126}a^{12}-\frac{97354163}{996726042}a^{11}+\frac{18627683}{1495089063}a^{10}+\frac{838148867}{2990178126}a^{9}+\frac{1074462953}{996726042}a^{8}+\frac{989137393}{1495089063}a^{7}-\frac{195992122}{115006851}a^{6}-\frac{319531255}{996726042}a^{5}-\frac{748177355}{996726042}a^{4}-\frac{241671327}{110747338}a^{3}-\frac{953020007}{1993452084}a^{2}+\frac{10465930775}{5980356252}a+\frac{118242322}{1495089063}$, $\frac{131703529}{2990178126}a^{14}-\frac{35219164}{1495089063}a^{13}-\frac{392762183}{2990178126}a^{12}-\frac{68638309}{2990178126}a^{11}+\frac{114116117}{2990178126}a^{10}+\frac{1638447419}{2990178126}a^{9}+\frac{5798055481}{2990178126}a^{8}-\frac{3490543157}{2990178126}a^{7}-\frac{7196096717}{2990178126}a^{6}+\frac{1299413593}{332242014}a^{5}-\frac{981695881}{110747338}a^{4}+\frac{5667238495}{996726042}a^{3}-\frac{5448715238}{1495089063}a^{2}+\frac{6018983855}{2990178126}a+\frac{1272996175}{1495089063}$, $\frac{12869807}{5980356252}a^{14}+\frac{102828013}{5980356252}a^{13}-\frac{1470855}{55373669}a^{12}-\frac{87303722}{1495089063}a^{11}+\frac{119867443}{2990178126}a^{10}+\frac{48049444}{498363021}a^{9}+\frac{381427070}{1495089063}a^{8}+\frac{1628137391}{2990178126}a^{7}-\frac{1167993683}{996726042}a^{6}-\frac{315247015}{498363021}a^{5}+\frac{1436637160}{498363021}a^{4}-\frac{1503018577}{498363021}a^{3}+\frac{8963206909}{5980356252}a^{2}-\frac{5858462743}{5980356252}a-\frac{286702826}{498363021}$
| 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: | \( 31837.7190213 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 31837.7190213 \cdot 1}{2\cdot\sqrt{30638157055420022784}}\cr\approx \mathstrut & 1.41563022959 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4)
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^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4, 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^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4);
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^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4);
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
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 non-solvable group of order 360 |
| The 7 conjugacy class representatives for $A_6$ |
| Character table for $A_6$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 6 siblings: | 6.2.7884864.1, 6.2.1971216.1 |
| Degree 10 sibling: | 10.2.15542770074624.1 |
| Degree 15 sibling: | deg 15 |
| Degree 20 sibling: | 20.4.241577701592627342528741376.1 |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 sibling: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | 6.2.1971216.1 |
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 | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{5}$ | ${\href{/padicField/19.3.0.1}{3} }^{5}$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.6.6.7 | $x^{6} + 2 x^{2} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
| 2.8.8.11 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(3\)
| 3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
| 3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |