Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28)
gp: K = bnfinit(y^15 - y^14 + 2*y^13 + 2*y^12 + 42*y^11 - 70*y^10 + 140*y^9 - 96*y^8 + 293*y^7 - 541*y^6 + 510*y^5 - 570*y^4 + 380*y^3 - 216*y^2 + 120*y - 28, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28)
\( x^{15} - x^{14} + 2 x^{13} + 2 x^{12} + 42 x^{11} - 70 x^{10} + 140 x^{9} - 96 x^{8} + 293 x^{7} + \cdots - 28 \)
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: |
\(-32552805874652777851904\)
\(\medspace = -\,2^{10}\cdot 13^{7}\cdot 47^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.68\) | 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: | $2^{2/3}13^{1/2}47^{1/2}\approx 39.23803668599558$ | ||
| Ramified primes: |
\(2\), \(13\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-611}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{52}a^{11}-\frac{3}{26}a^{10}+\frac{1}{26}a^{9}+\frac{1}{26}a^{8}+\frac{7}{52}a^{7}+\frac{5}{26}a^{6}+\frac{7}{26}a^{5}-\frac{3}{13}a^{4}+\frac{11}{26}a^{3}+\frac{6}{13}a^{2}-\frac{2}{13}a-\frac{2}{13}$, $\frac{1}{52}a^{12}+\frac{5}{52}a^{10}+\frac{1}{52}a^{9}-\frac{7}{52}a^{8}+\frac{9}{52}a^{6}+\frac{7}{52}a^{5}-\frac{6}{13}a^{4}-\frac{1}{2}a^{3}+\frac{3}{26}a^{2}-\frac{1}{13}a+\frac{1}{13}$, $\frac{1}{104}a^{13}-\frac{1}{104}a^{12}-\frac{1}{104}a^{11}-\frac{7}{104}a^{10}-\frac{7}{104}a^{9}-\frac{5}{104}a^{8}-\frac{7}{104}a^{7}+\frac{3}{104}a^{6}+\frac{1}{52}a^{5}-\frac{17}{52}a^{4}-\frac{6}{13}a^{3}+\frac{7}{26}a^{2}-\frac{6}{13}a+\frac{11}{26}$, $\frac{1}{8222466512}a^{14}+\frac{819025}{513904157}a^{13}+\frac{24053709}{4111233256}a^{12}+\frac{1075390}{513904157}a^{11}+\frac{100674219}{4111233256}a^{10}-\frac{22739503}{2055616628}a^{9}-\frac{121822951}{2055616628}a^{8}+\frac{47788381}{1027808314}a^{7}-\frac{183196447}{8222466512}a^{6}-\frac{8816433}{21868262}a^{5}-\frac{803350347}{4111233256}a^{4}-\frac{455112729}{1027808314}a^{3}-\frac{564686679}{2055616628}a^{2}-\frac{708477555}{2055616628}a+\frac{19592823}{2055616628}$
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: |
$\frac{160916417}{8222466512}a^{14}+\frac{11889479}{2055616628}a^{13}+\frac{55702149}{4111233256}a^{12}+\frac{159172803}{2055616628}a^{11}+\frac{3621924817}{4111233256}a^{10}-\frac{165023269}{513904157}a^{9}+\frac{1898996427}{2055616628}a^{8}+\frac{2363544851}{2055616628}a^{7}+\frac{33157916901}{8222466512}a^{6}-\frac{95408987}{21868262}a^{5}-\frac{13452692779}{4111233256}a^{4}-\frac{1261485141}{1027808314}a^{3}-\frac{1307399563}{2055616628}a^{2}+\frac{4484195137}{2055616628}a-\frac{977720981}{2055616628}$, $\frac{3953179}{357498544}a^{14}-\frac{3032647}{178749272}a^{13}+\frac{1179681}{44687318}a^{12}+\frac{3237405}{178749272}a^{11}+\frac{20092323}{44687318}a^{10}-\frac{181653065}{178749272}a^{9}+\frac{342288815}{178749272}a^{8}-\frac{263801069}{178749272}a^{7}+\frac{1213637445}{357498544}a^{6}-\frac{13255057}{1901588}a^{5}+\frac{1530983209}{178749272}a^{4}-\frac{144163969}{22343659}a^{3}+\frac{387066633}{89374636}a^{2}-\frac{117212773}{89374636}a+\frac{17196791}{89374636}$, $\frac{4174233}{8222466512}a^{14}-\frac{61604093}{2055616628}a^{13}+\frac{219493851}{4111233256}a^{12}-\frac{88302139}{1027808314}a^{11}-\frac{45342991}{4111233256}a^{10}-\frac{1239497317}{1027808314}a^{9}+\frac{1559446429}{513904157}a^{8}-\frac{6213682639}{1027808314}a^{7}+\frac{45608942909}{8222466512}a^{6}-\frac{215095461}{21868262}a^{5}+\frac{80794488789}{4111233256}a^{4}-\frac{27751055021}{1027808314}a^{3}+\frac{51620895465}{2055616628}a^{2}-\frac{28514879823}{2055616628}a+\frac{6253893679}{2055616628}$, $\frac{21268653}{4111233256}a^{14}-\frac{18344075}{4111233256}a^{13}+\frac{84661341}{4111233256}a^{12}+\frac{65460769}{4111233256}a^{11}+\frac{982163041}{4111233256}a^{10}-\frac{1133894001}{4111233256}a^{9}+\frac{4903847315}{4111233256}a^{8}-\frac{2015041257}{4111233256}a^{7}+\frac{5139618457}{2055616628}a^{6}-\frac{18557922}{10934131}a^{5}+\frac{5720789161}{1027808314}a^{4}-\frac{2328678630}{513904157}a^{3}+\frac{2180829761}{1027808314}a^{2}-\frac{2626094229}{1027808314}a+\frac{499716111}{513904157}$, $\frac{91961587}{8222466512}a^{14}+\frac{102344439}{4111233256}a^{13}+\frac{22517048}{513904157}a^{12}+\frac{288497071}{4111233256}a^{11}+\frac{1278369775}{2055616628}a^{10}+\frac{3806548743}{4111233256}a^{9}+\frac{6303880279}{4111233256}a^{8}+\frac{5572940265}{4111233256}a^{7}+\frac{45765707297}{8222466512}a^{6}+\frac{46399194}{10934131}a^{5}+\frac{121449749}{4111233256}a^{4}-\frac{8750363757}{1027808314}a^{3}-\frac{5084062549}{2055616628}a^{2}-\frac{6630744385}{2055616628}a+\frac{2832908399}{2055616628}$, $\frac{56691663}{1027808314}a^{14}-\frac{30142539}{4111233256}a^{13}+\frac{473773091}{4111233256}a^{12}+\frac{847936627}{4111233256}a^{11}+\frac{10356168317}{4111233256}a^{10}-\frac{6702477785}{4111233256}a^{9}+\frac{28072677563}{4111233256}a^{8}+\frac{829315197}{4111233256}a^{7}+\frac{73163568659}{4111233256}a^{6}-\frac{314183915}{21868262}a^{5}+\frac{39533520529}{2055616628}a^{4}-\frac{9305107938}{513904157}a^{3}+\frac{4856598326}{513904157}a^{2}-\frac{3391537615}{513904157}a+\frac{2496218567}{1027808314}$, $\frac{35790}{39531089}a^{14}+\frac{1017763}{24326824}a^{13}-\frac{20708075}{316248712}a^{12}+\frac{32059177}{316248712}a^{11}+\frac{33636571}{316248712}a^{10}+\frac{521668341}{316248712}a^{9}-\frac{1236529491}{316248712}a^{8}+\frac{2280542599}{316248712}a^{7}-\frac{1774928023}{316248712}a^{6}+\frac{19726305}{1682174}a^{5}-\frac{4350688701}{158124356}a^{4}+\frac{1207116238}{39531089}a^{3}-\frac{1006632258}{39531089}a^{2}+\frac{554554319}{39531089}a-\frac{255659193}{79062178}$
| 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: | \( 645647.809273 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 645647.809273 \cdot 1}{2\cdot\sqrt{32552805874652777851904}}\cr\approx \mathstrut & 1.38344069330 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28)
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 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28, 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 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28);
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 - x^14 + 2*x^13 + 2*x^12 + 42*x^11 - 70*x^10 + 140*x^9 - 96*x^8 + 293*x^7 - 541*x^6 + 510*x^5 - 570*x^4 + 380*x^3 - 216*x^2 + 120*x - 28);
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
| 3.1.2444.1, 5.1.373321.1 |
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 | $15$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15$ | R | $15$ | ${\href{/padicField/19.5.0.1}{5} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | R | $15$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |