Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*x - 4)
gp: K = bnfinit(y^15 - 3*y^14 - 4*y^13 + 6*y^12 + 20*y^11 - 8*y^10 + 10*y^9 + 50*y^8 - 45*y^7 - 65*y^6 + 106*y^5 + 80*y^4 - 20*y^3 - 40*y^2 - 20*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*x - 4)
\( x^{15} - 3 x^{14} - 4 x^{13} + 6 x^{12} + 20 x^{11} - 8 x^{10} + 10 x^{9} + 50 x^{8} - 45 x^{7} + \cdots - 4 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-31803942308779830272\)
\(\medspace = -\,2^{10}\cdot 227^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.96\) | 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: | $2^{2/3}227^{1/2}\approx 23.916608385226205$ | ||
| Ramified primes: |
\(2\), \(227\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-227}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{13}+\frac{1}{40}a^{12}+\frac{1}{40}a^{11}+\frac{1}{40}a^{10}-\frac{1}{8}a^{9}+\frac{3}{40}a^{8}+\frac{7}{40}a^{7}-\frac{9}{40}a^{6}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{10}a+\frac{1}{10}$, $\frac{1}{575759600}a^{14}-\frac{6021}{35984975}a^{13}+\frac{17045571}{143939900}a^{12}+\frac{30776117}{287879800}a^{11}-\frac{25558651}{287879800}a^{10}-\frac{22432671}{287879800}a^{9}+\frac{9194187}{71969950}a^{8}+\frac{50263591}{287879800}a^{7}-\frac{96431451}{575759600}a^{6}+\frac{40821409}{287879800}a^{5}+\frac{33303689}{71969950}a^{4}+\frac{28332371}{143939900}a^{3}-\frac{12940849}{71969950}a^{2}+\frac{3317787}{71969950}a-\frac{59624197}{143939900}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{11052087}{287879800}a^{14}-\frac{31191441}{143939900}a^{13}+\frac{16777077}{71969950}a^{12}+\frac{47203329}{143939900}a^{11}+\frac{5639263}{143939900}a^{10}-\frac{121036601}{71969950}a^{9}+\frac{349679451}{143939900}a^{8}-\frac{111048333}{143939900}a^{7}-\frac{1493533437}{287879800}a^{6}+\frac{715427683}{143939900}a^{5}+\frac{614518547}{143939900}a^{4}-\frac{642907823}{71969950}a^{3}+\frac{127294649}{71969950}a^{2}-\frac{42072337}{71969950}a+\frac{19400461}{71969950}$, $\frac{155342309}{575759600}a^{14}-\frac{129426051}{143939900}a^{13}-\frac{57278703}{71969950}a^{12}+\frac{557128963}{287879800}a^{11}+\frac{1382418201}{287879800}a^{10}-\frac{1113147339}{287879800}a^{9}+\frac{532747781}{143939900}a^{8}+\frac{3588480589}{287879800}a^{7}-\frac{9522962639}{575759600}a^{6}-\frac{3696464409}{287879800}a^{5}+\frac{1225000063}{35984975}a^{4}+\frac{1650835599}{143939900}a^{3}-\frac{407885248}{35984975}a^{2}-\frac{614112357}{71969950}a-\frac{270560153}{143939900}$, $\frac{26084381}{575759600}a^{14}-\frac{45849863}{287879800}a^{13}-\frac{36329853}{287879800}a^{12}+\frac{29010393}{71969950}a^{11}+\frac{61355041}{71969950}a^{10}-\frac{33340622}{35984975}a^{9}+\frac{109471723}{287879800}a^{8}+\frac{71079167}{35984975}a^{7}-\frac{1918634941}{575759600}a^{6}-\frac{874960551}{287879800}a^{5}+\frac{531891159}{71969950}a^{4}+\frac{263040371}{143939900}a^{3}-\frac{274015679}{71969950}a^{2}-\frac{71017029}{35984975}a-\frac{1145867}{143939900}$, $\frac{79554541}{287879800}a^{14}-\frac{32566252}{35984975}a^{13}-\frac{30333337}{35984975}a^{12}+\frac{263588827}{143939900}a^{11}+\frac{180530041}{35984975}a^{10}-\frac{490936161}{143939900}a^{9}+\frac{547955533}{143939900}a^{8}+\frac{1781283791}{143939900}a^{7}-\frac{4486396381}{287879800}a^{6}-\frac{1868442701}{143939900}a^{5}+\frac{4430732071}{143939900}a^{4}+\frac{1022413641}{71969950}a^{3}-\frac{521365973}{71969950}a^{2}-\frac{737483581}{71969950}a-\frac{254238617}{71969950}$, $\frac{57549711}{287879800}a^{14}-\frac{95870223}{143939900}a^{13}-\frac{21196497}{35984975}a^{12}+\frac{207515487}{143939900}a^{11}+\frac{503339839}{143939900}a^{10}-\frac{101097639}{35984975}a^{9}+\frac{404703803}{143939900}a^{8}+\frac{1337158301}{143939900}a^{7}-\frac{3610057661}{287879800}a^{6}-\frac{1241644401}{143939900}a^{5}+\frac{3597948941}{143939900}a^{4}+\frac{530452581}{71969950}a^{3}-\frac{524229853}{71969950}a^{2}-\frac{391343661}{71969950}a-\frac{98137567}{71969950}$, $\frac{95067521}{575759600}a^{14}-\frac{90929369}{143939900}a^{13}-\frac{7497541}{35984975}a^{12}+\frac{392249947}{287879800}a^{11}+\frac{704985369}{287879800}a^{10}-\frac{1048571241}{287879800}a^{9}+\frac{244943757}{71969950}a^{8}+\frac{1634687941}{287879800}a^{7}-\frac{7756086491}{575759600}a^{6}-\frac{892699871}{287879800}a^{5}+\frac{3186228713}{143939900}a^{4}-\frac{331804519}{143939900}a^{3}-\frac{273891012}{35984975}a^{2}-\frac{185867954}{35984975}a-\frac{241537557}{143939900}$, $\frac{11052087}{287879800}a^{14}-\frac{31191441}{143939900}a^{13}+\frac{16777077}{71969950}a^{12}+\frac{47203329}{143939900}a^{11}+\frac{5639263}{143939900}a^{10}-\frac{121036601}{71969950}a^{9}+\frac{349679451}{143939900}a^{8}-\frac{111048333}{143939900}a^{7}-\frac{1493533437}{287879800}a^{6}+\frac{715427683}{143939900}a^{5}+\frac{614518547}{143939900}a^{4}-\frac{642907823}{71969950}a^{3}+\frac{127294649}{71969950}a^{2}+\frac{29897613}{71969950}a+\frac{91370411}{71969950}$
| 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: | \( 13535.3893625 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 13535.3893625 \cdot 1}{2\cdot\sqrt{31803942308779830272}}\cr\approx \mathstrut & 0.927874743113 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*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^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*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^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*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^14 - 4*x^13 + 6*x^12 + 20*x^11 - 8*x^10 + 10*x^9 + 50*x^8 - 45*x^7 - 65*x^6 + 106*x^5 + 80*x^4 - 20*x^3 - 40*x^2 - 20*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 solvable group of order 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.908.1, 5.1.51529.1 |
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 | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15$ | $15$ | $15$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/59.3.0.1}{3} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(227\)
| $\Q_{227}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |