Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*x^2 + 6*x - 1)
gp: K = bnfinit(y^15 - 4*y^14 + 4*y^13 + 4*y^12 - 5*y^11 - 13*y^10 + 20*y^9 + 4*y^8 - 15*y^7 - 13*y^6 + 27*y^5 - 4*y^4 - 8*y^3 - 2*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*x^2 + 6*x - 1)
\( x^{15} - 4 x^{14} + 4 x^{13} + 4 x^{12} - 5 x^{11} - 13 x^{10} + 20 x^{9} + 4 x^{8} - 15 x^{7} + \cdots - 1 \)
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: |
\(-44543599279432079\)
\(\medspace = -\,239^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.88\) | 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: | $239^{1/2}\approx 15.459624833740307$ | ||
| Ramified primes: |
\(239\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-239}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{34333}a^{14}-\frac{6031}{34333}a^{13}-\frac{9806}{34333}a^{12}+\frac{13673}{34333}a^{11}-\frac{7976}{34333}a^{10}+\frac{5139}{34333}a^{9}-\frac{4367}{34333}a^{8}-\frac{13498}{34333}a^{7}-\frac{16779}{34333}a^{6}+\frac{16335}{34333}a^{5}+\frac{16026}{34333}a^{4}-\frac{9977}{34333}a^{3}+\frac{752}{1807}a^{2}-\frac{6614}{34333}a+\frac{1971}{34333}$
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: |
$\frac{2825}{34333}a^{14}-\frac{8407}{34333}a^{13}+\frac{4781}{34333}a^{12}+\frac{1600}{34333}a^{11}+\frac{24581}{34333}a^{10}-\frac{39517}{34333}a^{9}-\frac{11228}{34333}a^{8}+\frac{12113}{34333}a^{7}+\frac{81864}{34333}a^{6}-\frac{65843}{34333}a^{5}-\frac{46110}{34333}a^{4}+\frac{2368}{34333}a^{3}+\frac{4789}{1807}a^{2}-\frac{41731}{34333}a-\frac{28204}{34333}$, $\frac{9151}{34333}a^{14}-\frac{16550}{34333}a^{13}-\frac{22577}{34333}a^{12}+\frac{46504}{34333}a^{11}+\frac{72248}{34333}a^{10}-\frac{112220}{34333}a^{9}-\frac{101804}{34333}a^{8}+\frac{147268}{34333}a^{7}+\frac{129879}{34333}a^{6}-\frac{175962}{34333}a^{5}-\frac{119649}{34333}a^{4}+\frac{129252}{34333}a^{3}+\frac{4110}{1807}a^{2}-\frac{29968}{34333}a-\frac{22537}{34333}$, $\frac{4072}{34333}a^{14}-\frac{10137}{34333}a^{13}-\frac{753}{34333}a^{12}+\frac{22663}{34333}a^{11}+\frac{746}{34333}a^{10}-\frac{51455}{34333}a^{9}+\frac{36403}{34333}a^{8}+\frac{37610}{34333}a^{7}-\frac{35751}{34333}a^{6}-\frac{55567}{34333}a^{5}+\frac{59505}{34333}a^{4}-\frac{10405}{34333}a^{3}-\frac{721}{1807}a^{2}-\frac{15136}{34333}a+\frac{26323}{34333}$, $\frac{1179}{34333}a^{14}-\frac{3618}{34333}a^{13}+\frac{8947}{34333}a^{12}-\frac{16043}{34333}a^{11}+\frac{3538}{34333}a^{10}+\frac{16273}{34333}a^{9}+\frac{35590}{34333}a^{8}-\frac{86629}{34333}a^{7}-\frac{6633}{34333}a^{6}+\frac{66818}{34333}a^{5}+\frac{45837}{34333}a^{4}-\frac{123996}{34333}a^{3}+\frac{1178}{1807}a^{2}+\frac{30018}{34333}a+\frac{23498}{34333}$, $\frac{1744}{34333}a^{14}-\frac{12166}{34333}a^{13}+\frac{30503}{34333}a^{12}-\frac{15723}{34333}a^{11}-\frac{39612}{34333}a^{10}+\frac{1503}{34333}a^{9}+\frac{143210}{34333}a^{8}-\frac{91073}{34333}a^{7}-\frac{113859}{34333}a^{6}+\frac{26183}{34333}a^{5}+\frac{208280}{34333}a^{4}-\frac{130389}{34333}a^{3}-\frac{4008}{1807}a^{2}+\frac{1072}{34333}a+\frac{72790}{34333}$, $\frac{4567}{34333}a^{14}-\frac{8511}{34333}a^{13}-\frac{13770}{34333}a^{12}+\frac{27197}{34333}a^{11}+\frac{35254}{34333}a^{10}-\frac{48292}{34333}a^{9}-\frac{65282}{34333}a^{8}+\frac{51035}{34333}a^{7}+\frac{70229}{34333}a^{6}-\frac{37997}{34333}a^{5}-\frac{75880}{34333}a^{4}+\frac{29265}{34333}a^{3}+\frac{2891}{1807}a^{2}+\frac{6902}{34333}a-\frac{28022}{34333}$, $\frac{13063}{34333}a^{14}-\frac{57384}{34333}a^{13}+\frac{69311}{34333}a^{12}+\frac{44466}{34333}a^{11}-\frac{92832}{34333}a^{10}-\frac{161923}{34333}a^{9}+\frac{324322}{34333}a^{8}+\frac{9914}{34333}a^{7}-\frac{242536}{34333}a^{6}-\frac{132822}{34333}a^{5}+\frac{397000}{34333}a^{4}-\frac{104482}{34333}a^{3}-\frac{4897}{1807}a^{2}-\frac{16854}{34333}a+\frac{31756}{34333}$
| 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: | \( 124.657592501 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 124.657592501 \cdot 1}{2\cdot\sqrt{44543599279432079}}\cr\approx \mathstrut & 0.228341663351 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*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 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*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 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*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 - 4*x^14 + 4*x^13 + 4*x^12 - 5*x^11 - 13*x^10 + 20*x^9 + 4*x^8 - 15*x^7 - 13*x^6 + 27*x^5 - 4*x^4 - 8*x^3 - 2*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 solvable group of order 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.239.1, 5.1.57121.1 |
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
| Galois closure: | deg 30 |
| 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 | $15$ | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(239\)
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.239.2t1.a.a | $1$ | $ 239 $ | \(\Q(\sqrt{-239}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.239.3t2.a.a | $2$ | $ 239 $ | 3.1.239.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.239.5t2.a.a | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.239.5t2.a.b | $2$ | $ 239 $ | 5.1.57121.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.239.15t2.a.c | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.239.15t2.a.a | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.239.15t2.a.b | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.239.15t2.a.d | $2$ | $ 239 $ | 15.1.44543599279432079.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.