Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1)
gp: K = bnfinit(y^15 - 4*y^14 + 6*y^13 - y^12 - 11*y^11 + 18*y^10 - 10*y^9 - 9*y^8 + 19*y^7 - 12*y^6 - 3*y^5 + 9*y^4 - 5*y^3 - y^2 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1)
\( x^{15} - 4 x^{14} + 6 x^{13} - x^{12} - 11 x^{11} + 18 x^{10} - 10 x^{9} - 9 x^{8} + 19 x^{7} - 12 x^{6} + \cdots - 1 \)
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: | $[5, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10496055636998343\)
\(\medspace = -\,3^{5}\cdot 157\cdot 2377\cdot 115742009\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.70\) | 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: | $3^{1/2}157^{1/2}2377^{1/2}115742009^{1/2}\approx 11383362.147893872$ | ||
| Ramified primes: |
\(3\), \(157\), \(2377\), \(115742009\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-129580933790103}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $9$ | 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: |
$a$, $a^{14}-4a^{13}+6a^{12}-a^{11}-10a^{10}+15a^{9}-7a^{8}-9a^{7}+15a^{6}-7a^{5}-6a^{4}+8a^{3}-3a^{2}-2a+3$, $a^{14}-3a^{13}+2a^{12}+5a^{11}-12a^{10}+8a^{9}+5a^{8}-16a^{7}+11a^{6}+a^{5}-10a^{4}+6a^{3}-a^{2}-3a+2$, $a^{12}-4a^{11}+7a^{10}-5a^{9}-4a^{8}+12a^{7}-11a^{6}+a^{5}+5a^{4}-5a^{3}-a^{2}+2a-1$, $a^{13}-4a^{12}+6a^{11}-2a^{10}-7a^{9}+12a^{8}-8a^{7}-2a^{6}+7a^{5}-5a^{4}+a^{3}+a^{2}-a+1$, $a^{14}-4a^{13}+6a^{12}-2a^{11}-7a^{10}+11a^{9}-5a^{8}-5a^{7}+7a^{6}-a^{5}-4a^{4}+4a^{3}-a+1$, $a^{10}-3a^{9}+3a^{8}+a^{7}-6a^{6}+5a^{5}-5a^{3}+2a^{2}+a-2$, $a^{14}-3a^{13}+2a^{12}+6a^{11}-15a^{10}+11a^{9}+5a^{8}-20a^{7}+16a^{6}-2a^{5}-11a^{4}+7a^{3}-a^{2}-3a+3$, $a^{14}-3a^{13}+4a^{12}-2a^{11}-4a^{10}+10a^{9}-11a^{8}+9a^{6}-11a^{5}+a^{4}+4a^{3}-5a^{2}+2$
| 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: | \( 99.2990074596 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 99.2990074596 \cdot 1}{2\cdot\sqrt{10496055636998343}}\cr\approx \mathstrut & 0.151862718497 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*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 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*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 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*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 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*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 $S_{15}$ are not computed |
| Character table for $S_{15}$ is not computed |
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 | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | $15$ | $15$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{3}$ | $15$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(157\)
| 157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 157.13.0.1 | $x^{13} + 156 x^{2} + 9 x + 152$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(2377\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(115742009\)
| $\Q_{115742009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |