Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325)
gp: K = bnfinit(y^15 - 3*y^13 - 17*y^12 - 27*y^11 + 48*y^10 + 165*y^9 + 306*y^8 - 45*y^7 - 918*y^6 - 1251*y^5 - 633*y^4 + 2241*y^3 + 864*y^2 - 675*y - 325, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^13 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^13 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325)
\( x^{15} - 3 x^{13} - 17 x^{12} - 27 x^{11} + 48 x^{10} + 165 x^{9} + 306 x^{8} - 45 x^{7} - 918 x^{6} + \cdots - 325 \)
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: |
\(-1766485593616332297663\)
\(\medspace = -\,3^{20}\cdot 47^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.09\) | 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: | $3^{4/3}47^{1/2}\approx 29.66269470481324$ | ||
| Ramified primes: |
\(3\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{10585}a^{13}-\frac{934}{10585}a^{12}+\frac{1236}{10585}a^{11}+\frac{892}{2117}a^{10}-\frac{152}{365}a^{9}+\frac{3879}{10585}a^{8}-\frac{3476}{10585}a^{7}+\frac{5172}{10585}a^{6}+\frac{2851}{10585}a^{5}+\frac{763}{10585}a^{4}-\frac{143}{2117}a^{3}-\frac{161}{10585}a^{2}+\frac{2042}{10585}a-\frac{72}{2117}$, $\frac{1}{15\!\cdots\!65}a^{14}+\frac{401251463169}{15\!\cdots\!65}a^{13}+\frac{10307305984552}{207948175783205}a^{12}-\frac{45\!\cdots\!38}{15\!\cdots\!65}a^{11}-\frac{369222989661827}{15\!\cdots\!65}a^{10}-\frac{476935187284568}{11\!\cdots\!05}a^{9}-\frac{68\!\cdots\!44}{15\!\cdots\!65}a^{8}-\frac{60\!\cdots\!33}{15\!\cdots\!65}a^{7}+\frac{40\!\cdots\!03}{15\!\cdots\!65}a^{6}-\frac{46\!\cdots\!44}{15\!\cdots\!65}a^{5}-\frac{26\!\cdots\!64}{15\!\cdots\!65}a^{4}-\frac{62\!\cdots\!78}{15\!\cdots\!65}a^{3}-\frac{192100935619007}{798958780640735}a^{2}-\frac{49\!\cdots\!68}{15\!\cdots\!65}a+\frac{79799278342637}{233541797418061}$
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
$C_{3}$, which has order $3$
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: |
$\frac{1592829194301}{13\!\cdots\!15}a^{14}+\frac{4384558055916}{13\!\cdots\!15}a^{13}-\frac{2016727510856}{13\!\cdots\!15}a^{12}-\frac{38832168663954}{13\!\cdots\!15}a^{11}-\frac{125200191811059}{13\!\cdots\!15}a^{10}-\frac{7221393526269}{106155362462755}a^{9}+\frac{371330561726109}{13\!\cdots\!15}a^{8}+\frac{12\!\cdots\!74}{13\!\cdots\!15}a^{7}+\frac{356941889703008}{276003942403163}a^{6}-\frac{508689902277252}{13\!\cdots\!15}a^{5}-\frac{53\!\cdots\!82}{13\!\cdots\!15}a^{4}-\frac{87\!\cdots\!09}{13\!\cdots\!15}a^{3}-\frac{215638712043579}{72632616421885}a^{2}+\frac{56\!\cdots\!69}{13\!\cdots\!15}a+\frac{70984366789457}{21231072492551}$, $\frac{5629831431534}{13\!\cdots\!15}a^{14}+\frac{10688696037}{276003942403163}a^{13}-\frac{15334379787898}{13\!\cdots\!15}a^{12}-\frac{19491964826706}{276003942403163}a^{11}-\frac{160681967184501}{13\!\cdots\!15}a^{10}+\frac{18560633336458}{106155362462755}a^{9}+\frac{180380918840238}{276003942403163}a^{8}+\frac{377659362551283}{276003942403163}a^{7}+\frac{101615365058747}{13\!\cdots\!15}a^{6}-\frac{46\!\cdots\!02}{13\!\cdots\!15}a^{5}-\frac{14\!\cdots\!80}{276003942403163}a^{4}-\frac{58\!\cdots\!61}{13\!\cdots\!15}a^{3}+\frac{103089381921666}{14526523284377}a^{2}+\frac{28\!\cdots\!88}{13\!\cdots\!15}a-\frac{9255597295505}{21231072492551}$, $\frac{51337443435119}{15\!\cdots\!65}a^{14}+\frac{46006182354944}{15\!\cdots\!65}a^{13}-\frac{22266481186831}{30\!\cdots\!93}a^{12}-\frac{995878450915278}{15\!\cdots\!65}a^{11}-\frac{22\!\cdots\!84}{15\!\cdots\!65}a^{10}+\frac{25336849483533}{11\!\cdots\!05}a^{9}+\frac{92\!\cdots\!46}{15\!\cdots\!65}a^{8}+\frac{24\!\cdots\!02}{15\!\cdots\!65}a^{7}+\frac{18\!\cdots\!37}{15\!\cdots\!65}a^{6}-\frac{30\!\cdots\!48}{15\!\cdots\!65}a^{5}-\frac{99\!\cdots\!09}{15\!\cdots\!65}a^{4}-\frac{23\!\cdots\!47}{30\!\cdots\!93}a^{3}+\frac{790551326474543}{798958780640735}a^{2}+\frac{85\!\cdots\!23}{15\!\cdots\!65}a+\frac{432203516378709}{233541797418061}$, $\frac{14425986153171}{15\!\cdots\!65}a^{14}+\frac{5732595271374}{15\!\cdots\!65}a^{13}-\frac{44574101657571}{15\!\cdots\!65}a^{12}-\frac{222099252467452}{15\!\cdots\!65}a^{11}-\frac{475040108446933}{15\!\cdots\!65}a^{10}+\frac{5067587682122}{233541797418061}a^{9}+\frac{18\!\cdots\!86}{15\!\cdots\!65}a^{8}+\frac{46\!\cdots\!14}{15\!\cdots\!65}a^{7}+\frac{36\!\cdots\!22}{15\!\cdots\!65}a^{6}-\frac{61\!\cdots\!79}{15\!\cdots\!65}a^{5}-\frac{17\!\cdots\!51}{15\!\cdots\!65}a^{4}-\frac{24\!\cdots\!41}{15\!\cdots\!65}a^{3}+\frac{467080072457006}{798958780640735}a^{2}+\frac{93\!\cdots\!96}{15\!\cdots\!65}a+\frac{27765319806118}{233541797418061}$, $\frac{70124683405058}{15\!\cdots\!65}a^{14}-\frac{4846019687211}{30\!\cdots\!93}a^{13}-\frac{197887360294797}{15\!\cdots\!65}a^{12}-\frac{11\!\cdots\!02}{15\!\cdots\!65}a^{11}-\frac{288016845760346}{30\!\cdots\!93}a^{10}+\frac{58310457482676}{233541797418061}a^{9}+\frac{19\!\cdots\!10}{30\!\cdots\!93}a^{8}+\frac{240677027470307}{207948175783205}a^{7}-\frac{95\!\cdots\!09}{15\!\cdots\!65}a^{6}-\frac{56\!\cdots\!24}{15\!\cdots\!65}a^{5}-\frac{62\!\cdots\!46}{15\!\cdots\!65}a^{4}-\frac{25\!\cdots\!41}{15\!\cdots\!65}a^{3}+\frac{82\!\cdots\!34}{798958780640735}a^{2}-\frac{11\!\cdots\!02}{15\!\cdots\!65}a-\frac{575822207787195}{233541797418061}$, $\frac{18041822402798}{15\!\cdots\!65}a^{14}+\frac{12814907788347}{15\!\cdots\!65}a^{13}-\frac{55389860715593}{15\!\cdots\!65}a^{12}-\frac{309921207910841}{15\!\cdots\!65}a^{11}-\frac{683309352570609}{15\!\cdots\!65}a^{10}+\frac{10035693699034}{233541797418061}a^{9}+\frac{29\!\cdots\!08}{15\!\cdots\!65}a^{8}+\frac{62\!\cdots\!32}{15\!\cdots\!65}a^{7}+\frac{11\!\cdots\!71}{15\!\cdots\!65}a^{6}-\frac{15\!\cdots\!47}{15\!\cdots\!65}a^{5}-\frac{19\!\cdots\!38}{15\!\cdots\!65}a^{4}-\frac{11\!\cdots\!73}{15\!\cdots\!65}a^{3}+\frac{21\!\cdots\!38}{798958780640735}a^{2}+\frac{87\!\cdots\!68}{15\!\cdots\!65}a-\frac{262966640746893}{233541797418061}$, $\frac{2574156898089}{13\!\cdots\!15}a^{14}+\frac{18602755734}{18904379616655}a^{13}-\frac{8672378031411}{13\!\cdots\!15}a^{12}-\frac{39576550709678}{13\!\cdots\!15}a^{11}-\frac{63048975287656}{13\!\cdots\!15}a^{10}+\frac{6342408229191}{106155362462755}a^{9}+\frac{348090147431623}{13\!\cdots\!15}a^{8}+\frac{593792378688893}{13\!\cdots\!15}a^{7}-\frac{112933679453149}{13\!\cdots\!15}a^{6}-\frac{328014209332058}{276003942403163}a^{5}-\frac{18\!\cdots\!34}{13\!\cdots\!15}a^{4}-\frac{367482265853981}{13\!\cdots\!15}a^{3}+\frac{189842015423577}{72632616421885}a^{2}+\frac{327151056783127}{13\!\cdots\!15}a-\frac{15115701958680}{21231072492551}$
| 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: | \( 14560.9805563 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 14560.9805563 \cdot 3}{2\cdot\sqrt{1766485593616332297663}}\cr\approx \mathstrut & 0.401805525934 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325)
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 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325, 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 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325);
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 - 17*x^12 - 27*x^11 + 48*x^10 + 165*x^9 + 306*x^8 - 45*x^7 - 918*x^6 - 1251*x^5 - 633*x^4 + 2241*x^3 + 864*x^2 - 675*x - 325);
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
| 3.1.3807.1, 5.1.2209.1 |
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 | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $15$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | R | $15$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.3807.3t2.a.a | $2$ | $ 3^{4} \cdot 47 $ | 3.1.3807.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.47.5t2.a.b | $2$ | $ 47 $ | 5.1.2209.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.47.5t2.a.a | $2$ | $ 47 $ | 5.1.2209.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.3807.15t2.a.a | $2$ | $ 3^{4} \cdot 47 $ | 15.1.1766485593616332297663.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3807.15t2.a.b | $2$ | $ 3^{4} \cdot 47 $ | 15.1.1766485593616332297663.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3807.15t2.a.c | $2$ | $ 3^{4} \cdot 47 $ | 15.1.1766485593616332297663.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.3807.15t2.a.d | $2$ | $ 3^{4} \cdot 47 $ | 15.1.1766485593616332297663.1 | $D_{15}$ (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 *.