Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225)
gp: K = bnfinit(y^15 - 4*y^14 + 3*y^13 + 8*y^12 + 27*y^11 - 92*y^10 + 113*y^9 + 8*y^8 + 334*y^7 - 96*y^6 + 2102*y^5 - 1080*y^4 + 2129*y^3 - 773*y^2 + 570*y - 225, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225)
\( x^{15} - 4 x^{14} + 3 x^{13} + 8 x^{12} + 27 x^{11} - 92 x^{10} + 113 x^{9} + 8 x^{8} + 334 x^{7} + \cdots - 225 \)
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: |
\(-34740892637448264404263\)
\(\medspace = -\,7^{10}\cdot 103^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.82\) | 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: | $7^{2/3}103^{1/2}\approx 37.13789685454593$ | ||
| Ramified primes: |
\(7\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-103}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{9}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{15}a$, $\frac{1}{15}a^{10}-\frac{2}{15}a^{8}+\frac{1}{15}a^{7}-\frac{1}{3}a^{6}-\frac{2}{15}a^{5}-\frac{2}{15}a^{4}-\frac{1}{3}a^{3}-\frac{2}{5}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{11}+\frac{1}{15}a^{8}+\frac{1}{15}a^{7}-\frac{1}{3}a^{6}-\frac{2}{15}a^{5}+\frac{1}{15}a^{4}-\frac{1}{3}a^{2}-\frac{2}{15}a$, $\frac{1}{45}a^{12}-\frac{1}{45}a^{11}+\frac{1}{45}a^{9}-\frac{1}{9}a^{8}-\frac{11}{45}a^{7}-\frac{2}{45}a^{6}-\frac{2}{45}a^{5}-\frac{7}{15}a^{4}+\frac{1}{9}a^{3}+\frac{13}{45}a^{2}+\frac{4}{15}a$, $\frac{1}{458865}a^{13}-\frac{301}{152955}a^{12}+\frac{1406}{458865}a^{11}+\frac{8716}{458865}a^{10}+\frac{4652}{458865}a^{9}-\frac{76432}{458865}a^{8}+\frac{56456}{458865}a^{7}-\frac{167194}{458865}a^{6}+\frac{43664}{91773}a^{5}+\frac{16843}{41715}a^{4}-\frac{4}{99}a^{3}+\frac{104461}{458865}a^{2}+\frac{72163}{152955}a+\frac{1906}{10197}$, $\frac{1}{2085223431555}a^{14}+\frac{293780}{417044686311}a^{13}+\frac{2662008550}{417044686311}a^{12}+\frac{825215059}{695074477185}a^{11}-\frac{1662178274}{139014895437}a^{10}+\frac{10776400004}{417044686311}a^{9}-\frac{235308679919}{2085223431555}a^{8}-\frac{664871437094}{2085223431555}a^{7}+\frac{3057969838}{12637717767}a^{6}-\frac{8830938992}{46338298479}a^{5}+\frac{45854422466}{2085223431555}a^{4}-\frac{192370752688}{417044686311}a^{3}-\frac{836694004739}{2085223431555}a^{2}+\frac{202831342741}{695074477185}a-\frac{8758935170}{46338298479}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{193014263}{139014895437}a^{14}-\frac{2621797067}{695074477185}a^{13}-\frac{2698570279}{695074477185}a^{12}+\frac{4648886798}{231691492395}a^{11}+\frac{2302431695}{46338298479}a^{10}-\frac{62053839338}{695074477185}a^{9}-\frac{20775840584}{695074477185}a^{8}+\frac{204491757904}{695074477185}a^{7}+\frac{94471555133}{231691492395}a^{6}+\frac{7966204498}{21062862945}a^{5}+\frac{1753667110574}{695074477185}a^{4}+\frac{1520879513038}{695074477185}a^{3}-\frac{335137914494}{695074477185}a^{2}+\frac{735307900018}{231691492395}a-\frac{884100844}{1404190863}$, $\frac{730707682}{189565766505}a^{14}-\frac{6250176371}{417044686311}a^{13}+\frac{18193749184}{2085223431555}a^{12}+\frac{5019825352}{139014895437}a^{11}+\frac{73922982796}{695074477185}a^{10}-\frac{739308775573}{2085223431555}a^{9}+\frac{761254369856}{2085223431555}a^{8}+\frac{70390115965}{417044686311}a^{7}+\frac{168409989574}{139014895437}a^{6}-\frac{2717995903}{8581166385}a^{5}+\frac{1470876995576}{189565766505}a^{4}-\frac{6679072227391}{2085223431555}a^{3}+\frac{11487875763191}{2085223431555}a^{2}-\frac{1396265909518}{695074477185}a+\frac{15255141785}{46338298479}$, $\frac{6185458067}{2085223431555}a^{14}-\frac{3120799349}{189565766505}a^{13}+\frac{49753957999}{2085223431555}a^{12}+\frac{16906329911}{695074477185}a^{11}+\frac{24585852772}{695074477185}a^{10}-\frac{184290917786}{417044686311}a^{9}+\frac{25664609957}{37913153301}a^{8}-\frac{170308668571}{2085223431555}a^{7}+\frac{382669657856}{695074477185}a^{6}-\frac{477990467461}{231691492395}a^{5}+\frac{11606306454736}{2085223431555}a^{4}-\frac{2091208193717}{189565766505}a^{3}+\frac{7421815651934}{2085223431555}a^{2}-\frac{2863829715979}{695074477185}a+\frac{78830429504}{46338298479}$, $\frac{3357906143}{417044686311}a^{14}-\frac{5434727221}{189565766505}a^{13}+\frac{3174681208}{417044686311}a^{12}+\frac{11404288979}{139014895437}a^{11}+\frac{174249764426}{695074477185}a^{10}-\frac{1404555335408}{2085223431555}a^{9}+\frac{91667800502}{189565766505}a^{8}+\frac{264172421702}{417044686311}a^{7}+\frac{1928561556364}{695074477185}a^{6}+\frac{8448482036}{77230497465}a^{5}+\frac{32628508048736}{2085223431555}a^{4}-\frac{283695545734}{189565766505}a^{3}+\frac{19037186952391}{2085223431555}a^{2}-\frac{749313433271}{695074477185}a+\frac{81700455301}{46338298479}$, $\frac{118088768}{695074477185}a^{14}-\frac{243818789}{139014895437}a^{13}+\frac{3174465697}{695074477185}a^{12}-\frac{344863348}{231691492395}a^{11}-\frac{332154557}{231691492395}a^{10}-\frac{33961019614}{695074477185}a^{9}+\frac{63381069632}{695074477185}a^{8}-\frac{63574307056}{695074477185}a^{7}+\frac{5335141084}{46338298479}a^{6}-\frac{32452506734}{77230497465}a^{5}+\frac{62855634592}{695074477185}a^{4}-\frac{1853118699043}{695074477185}a^{3}+\frac{188170978660}{139014895437}a^{2}-\frac{55776745175}{46338298479}a+\frac{5677422419}{15446099493}$, $\frac{7152894884}{2085223431555}a^{14}-\frac{29222546236}{2085223431555}a^{13}+\frac{22450164523}{2085223431555}a^{12}+\frac{21977169017}{695074477185}a^{11}+\frac{57434259382}{695074477185}a^{10}-\frac{64310586749}{189565766505}a^{9}+\frac{880656985136}{2085223431555}a^{8}+\frac{401513298893}{2085223431555}a^{7}+\frac{634559720114}{695074477185}a^{6}-\frac{55782938326}{77230497465}a^{5}+\frac{3009762558569}{417044686311}a^{4}-\frac{5184411355741}{2085223431555}a^{3}+\frac{1316988957271}{189565766505}a^{2}-\frac{34141224266}{63188588835}a+\frac{85245416165}{46338298479}$, $\frac{550820293}{417044686311}a^{14}-\frac{12942771526}{2085223431555}a^{13}+\frac{15973540813}{2085223431555}a^{12}+\frac{4843751009}{695074477185}a^{11}+\frac{20391601363}{695074477185}a^{10}-\frac{27433428854}{189565766505}a^{9}+\frac{95890403167}{417044686311}a^{8}-\frac{181728584824}{2085223431555}a^{7}+\frac{231186126719}{695074477185}a^{6}-\frac{85704313819}{231691492395}a^{5}+\frac{5190586286674}{2085223431555}a^{4}-\frac{6432382439611}{2085223431555}a^{3}+\frac{531143267884}{189565766505}a^{2}-\frac{110030476091}{63188588835}a+\frac{11182688567}{46338298479}$
| 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: | \( 147159.120328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 147159.120328 \cdot 3}{2\cdot\sqrt{34740892637448264404263}}\cr\approx \mathstrut & 0.915686960402 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225)
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 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225, 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 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225);
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 + 3*x^13 + 8*x^12 + 27*x^11 - 92*x^10 + 113*x^9 + 8*x^8 + 334*x^7 - 96*x^6 + 2102*x^5 - 1080*x^4 + 2129*x^3 - 773*x^2 + 570*x - 225);
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
| 3.1.5047.1, 5.1.10609.1 |
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 | $15$ | ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.15.10.3 | $x^{15} + 98 x^{9} + 2401 x^{3} + 268912$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
|
\(103\)
| $\Q_{103}$ | $x + 98$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.103.2t1.a.a | $1$ | $ 103 $ | \(\Q(\sqrt{-103}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.5047.3t2.a.a | $2$ | $ 7^{2} \cdot 103 $ | 3.1.5047.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.103.5t2.a.b | $2$ | $ 103 $ | 5.1.10609.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.103.5t2.a.a | $2$ | $ 103 $ | 5.1.10609.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.5047.15t2.a.a | $2$ | $ 7^{2} \cdot 103 $ | 15.1.34740892637448264404263.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.5047.15t2.a.c | $2$ | $ 7^{2} \cdot 103 $ | 15.1.34740892637448264404263.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.5047.15t2.a.d | $2$ | $ 7^{2} \cdot 103 $ | 15.1.34740892637448264404263.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
| * | 2.5047.15t2.a.b | $2$ | $ 7^{2} \cdot 103 $ | 15.1.34740892637448264404263.1 | $D_{15}$ (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 *.