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Properties

Label 15.3.274502366303440896.1
Degree $15$
Signature $[3, 6]$
Discriminant $2.745\times 10^{17}$
Root discriminant \(14.54\)
Ramified primes $2,3,23$
Class number $1$
Class group trivial
Galois group $S_5 \times S_3$ (as 15T29)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^15 - y^14 - y^13 - 5*y^12 + 5*y^11 - 5*y^10 - y^9 + 3*y^8 + 7*y^7 + y^6 - 3*y^5 + y^4 - 13*y^3 + 5*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*x + 1)
 

\( x^{15} - x^{14} - x^{13} - 5 x^{12} + 5 x^{11} - 5 x^{10} - x^{9} + 3 x^{8} + 7 x^{7} + x^{6} - 3 x^{5} + \cdots + 1 \) Copy content Toggle raw display

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:  $[3, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(274502366303440896\) \(\medspace = 2^{12}\cdot 3^{9}\cdot 23^{7}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.54\)
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:  $2^{4/5}3^{7/6}23^{1/2}\approx 30.08357041040144$
Ramified primes:   \(2\), \(3\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{69}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{20}a^{13}+\frac{1}{10}a^{11}+\frac{1}{10}a^{10}+\frac{3}{20}a^{9}+\frac{1}{5}a^{8}+\frac{1}{10}a^{7}+\frac{1}{10}a^{6}-\frac{1}{4}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{4}a+\frac{2}{5}$, $\frac{1}{6740}a^{14}-\frac{17}{3370}a^{13}+\frac{447}{6740}a^{12}+\frac{409}{6740}a^{11}+\frac{65}{1348}a^{10}-\frac{283}{6740}a^{9}-\frac{1109}{6740}a^{8}+\frac{51}{1685}a^{7}-\frac{1333}{6740}a^{6}-\frac{292}{1685}a^{5}+\frac{1471}{6740}a^{4}-\frac{205}{1348}a^{3}+\frac{2471}{6740}a^{2}+\frac{353}{6740}a+\frac{143}{6740}$ Copy content Toggle raw display

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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{2167}{6740}a^{14}-\frac{53}{1685}a^{13}-\frac{899}{1685}a^{12}-\frac{1315}{674}a^{11}+\frac{141}{3370}a^{10}-\frac{907}{1685}a^{9}-\frac{9829}{6740}a^{8}+\frac{131}{337}a^{7}+\frac{18351}{6740}a^{6}+\frac{4166}{1685}a^{5}-\frac{13}{3370}a^{4}-\frac{2363}{3370}a^{3}-\frac{7398}{1685}a^{2}-\frac{2116}{1685}a-\frac{2181}{6740}$, $a$, $\frac{301}{3370}a^{14}+\frac{213}{3370}a^{13}-\frac{253}{3370}a^{12}-\frac{1296}{1685}a^{11}-\frac{3517}{6740}a^{10}-\frac{1607}{3370}a^{9}+\frac{653}{6740}a^{8}-\frac{976}{1685}a^{7}+\frac{1078}{1685}a^{6}+\frac{2826}{1685}a^{5}+\frac{1069}{674}a^{4}-\frac{1096}{1685}a^{3}-\frac{16489}{6740}a^{2}-\frac{1636}{1685}a-\frac{7937}{6740}$, $\frac{386}{1685}a^{14}+\frac{1647}{3370}a^{13}+\frac{341}{3370}a^{12}+\frac{238}{337}a^{11}-\frac{8597}{3370}a^{10}+\frac{6503}{3370}a^{9}-\frac{253}{1685}a^{8}-\frac{1235}{674}a^{7}-\frac{2083}{1685}a^{6}+\frac{6961}{3370}a^{5}+\frac{2398}{1685}a^{4}-\frac{6379}{3370}a^{3}+\frac{8907}{3370}a^{2}-\frac{14711}{3370}a+\frac{1151}{3370}$, $\frac{1086}{1685}a^{14}+\frac{2449}{6740}a^{13}+\frac{1523}{1685}a^{12}+\frac{4779}{1348}a^{11}-\frac{12239}{6740}a^{10}+\frac{3364}{1685}a^{9}+\frac{12213}{6740}a^{8}-\frac{1065}{674}a^{7}-\frac{20111}{3370}a^{6}-\frac{16593}{6740}a^{5}+\frac{2233}{1685}a^{4}+\frac{1167}{6740}a^{3}+\frac{53009}{6740}a^{2}+\frac{2507}{1685}a+\frac{7987}{6740}$, $\frac{229}{6740}a^{14}-\frac{709}{6740}a^{13}-\frac{211}{3370}a^{12}-\frac{5}{1348}a^{11}+\frac{4329}{6740}a^{10}+\frac{117}{3370}a^{9}-\frac{387}{1685}a^{8}+\frac{179}{337}a^{7}-\frac{1283}{6740}a^{6}-\frac{2927}{6740}a^{5}-\frac{1973}{1685}a^{4}-\frac{847}{6740}a^{3}+\frac{1721}{6740}a^{2}+\frac{4191}{3370}a+\frac{3399}{3370}$, $\frac{729}{1348}a^{14}+\frac{261}{674}a^{13}+\frac{1027}{1348}a^{12}+\frac{3791}{1348}a^{11}-\frac{1355}{674}a^{10}+\frac{2085}{1348}a^{9}+\frac{505}{337}a^{8}-\frac{1229}{674}a^{7}-\frac{5543}{1348}a^{6}-\frac{453}{337}a^{5}+\frac{3345}{1348}a^{4}-\frac{241}{1348}a^{3}+\frac{4671}{674}a^{2}-\frac{1891}{1348}a+\frac{140}{337}$, $\frac{4469}{6740}a^{14}-\frac{97}{674}a^{13}-\frac{2298}{1685}a^{12}-\frac{12671}{3370}a^{11}+\frac{1758}{1685}a^{10}+\frac{514}{1685}a^{9}-\frac{25137}{6740}a^{8}+\frac{5269}{3370}a^{7}+\frac{9363}{1348}a^{6}+\frac{13649}{3370}a^{5}-\frac{10427}{3370}a^{4}-\frac{3004}{1685}a^{3}-\frac{2338}{337}a^{2}-\frac{3692}{1685}a+\frac{2719}{1348}$ Copy content Toggle raw display
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:  \( 525.2779714178208 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 525.2779714178208 \cdot 1}{2\cdot\sqrt{274502366303440896}}\cr\approx \mathstrut & 0.246748967098216 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*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 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*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 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*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 - x^14 - x^13 - 5*x^12 + 5*x^11 - 5*x^10 - x^9 + 3*x^8 + 7*x^7 + x^6 - 3*x^5 + x^4 - 13*x^3 + 5*x^2 - 3*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

$S_3\times S_5$ (as 15T29):

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 non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$

Intermediate fields

3.1.23.1, 5.3.228528.1

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

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: data not computed
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 R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $15$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ R ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{5}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.15.12.1$x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1$$5$$3$$12$$F_5\times C_3$$[\ ]_{5}^{12}$
\(3\) Copy content Toggle raw display 3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.9.9.1$x^{9} + 90 x^{7} - 207 x^{6} + 540 x^{5} + 324 x^{4} + 243 x^{3} + 324 x^{2} + 162 x + 27$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.2$x^{4} - 483 x^{2} + 2645$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$