Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*x - 1)
gp: K = bnfinit(y^15 - 2*y^14 + 5*y^13 - 10*y^12 + 13*y^11 - 20*y^10 + 23*y^9 - 22*y^8 + 22*y^7 - 20*y^6 + 18*y^5 - 20*y^4 + 21*y^3 - 15*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*x - 1)
\( x^{15} - 2 x^{14} + 5 x^{13} - 10 x^{12} + 13 x^{11} - 20 x^{10} + 23 x^{9} - 22 x^{8} + 22 x^{7} + \cdots - 1 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-5976545641547631\)
\(\medspace = -\,3\cdot 195479\cdot 10191283363\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.27\) | 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: | $3^{1/2}195479^{1/2}10191283363^{1/2}\approx 77308121.44624671$ | ||
| Ramified primes: |
\(3\), \(195479\), \(10191283363\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-59765\!\cdots\!47631}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{61}a^{14}+\frac{28}{61}a^{13}-\frac{9}{61}a^{12}+\frac{25}{61}a^{11}-\frac{30}{61}a^{10}-\frac{5}{61}a^{9}-\frac{5}{61}a^{8}+\frac{11}{61}a^{7}-\frac{14}{61}a^{6}-\frac{13}{61}a^{5}-\frac{6}{61}a^{4}-\frac{17}{61}a^{3}-\frac{1}{61}a^{2}+\frac{16}{61}a-\frac{2}{61}$
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: | $7$ | 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: |
$a$, $\frac{524}{61}a^{14}-\frac{578}{61}a^{13}+\frac{2055}{61}a^{12}-\frac{3370}{61}a^{11}+\frac{3617}{61}a^{10}-\frac{7012}{61}a^{9}+\frac{5554}{61}a^{8}-\frac{6009}{61}a^{7}+\frac{5901}{61}a^{6}-\frac{4799}{61}a^{5}+\frac{4786}{61}a^{4}-\frac{5980}{61}a^{3}+\frac{5393}{61}a^{2}-\frac{2596}{61}a+\frac{477}{61}$, $\frac{485}{61}a^{14}-\frac{755}{61}a^{13}+\frac{2101}{61}a^{12}-\frac{3918}{61}a^{11}+\frac{4604}{61}a^{10}-\frac{7671}{61}a^{9}+\frac{7762}{61}a^{8}-\frac{7292}{61}a^{7}+\frac{7362}{61}a^{6}-\frac{6488}{61}a^{5}+\frac{5813}{61}a^{4}-\frac{7086}{61}a^{3}+\frac{7079}{61}a^{2}-\frac{4135}{61}a+\frac{1043}{61}$, $\frac{128}{61}a^{14}-\frac{76}{61}a^{13}+\frac{434}{61}a^{12}-\frac{582}{61}a^{11}+\frac{491}{61}a^{10}-\frac{1311}{61}a^{9}+\frac{580}{61}a^{8}-\frac{849}{61}a^{7}+\frac{831}{61}a^{6}-\frac{566}{61}a^{5}+\frac{574}{61}a^{4}-\frac{1017}{61}a^{3}+\frac{604}{61}a^{2}-\frac{26}{61}a-\frac{73}{61}$, $\frac{1305}{61}a^{14}-\frac{1890}{61}a^{13}+\frac{5457}{61}a^{12}-\frac{10014}{61}a^{11}+\frac{11358}{61}a^{10}-\frac{19701}{61}a^{9}+\frac{19034}{61}a^{8}-\frac{17975}{61}a^{7}+\frac{18635}{61}a^{6}-\frac{15684}{61}a^{5}+\frac{14679}{61}a^{4}-\frac{17854}{61}a^{3}+\frac{17422}{61}a^{2}-\frac{9742}{61}a+\frac{2270}{61}$, $\frac{2073}{61}a^{14}-\frac{2895}{61}a^{13}+\frac{8610}{61}a^{12}-\frac{15519}{61}a^{11}+\frac{17537}{61}a^{10}-\frac{30800}{61}a^{9}+\frac{28980}{61}a^{8}-\frac{27949}{61}a^{7}+\frac{28562}{61}a^{6}-\frac{24021}{61}a^{5}+\frac{22637}{61}a^{4}-\frac{27616}{61}a^{3}+\frac{26719}{61}a^{2}-\frac{14778}{61}a+\frac{3357}{61}$, $\frac{133}{61}a^{14}-\frac{58}{61}a^{13}+\frac{450}{61}a^{12}-\frac{518}{61}a^{11}+\frac{463}{61}a^{10}-\frac{1275}{61}a^{9}+\frac{372}{61}a^{8}-\frac{916}{61}a^{7}+\frac{578}{61}a^{6}-\frac{570}{61}a^{5}+\frac{544}{61}a^{4}-\frac{858}{61}a^{3}+\frac{538}{61}a^{2}-\frac{7}{61}a-\frac{144}{61}$
| 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: | \( 50.8430332143 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 50.8430332143 \cdot 1}{2\cdot\sqrt{5976545641547631}}\cr\approx \mathstrut & 0.254252604398 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*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 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*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 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*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 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*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
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 1307674368000 |
| The 176 conjugacy class representatives for $S_{15}$ |
| Character table for $S_{15}$ |
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 30 sibling: | 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 | $15$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $15$ | $15$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $15$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $15$ | $15$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.9.0.1 | $x^{9} + 2 x^{3} + 2 x^{2} + x + 1$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(195479\)
| $\Q_{195479}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{195479}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{195479}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(10191283363\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |