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Properties

Label 15.3.126064044311049216.1
Degree $15$
Signature $[3, 6]$
Discriminant $1.261\times 10^{17}$
Root discriminant \(13.81\)
Ramified primes $2,3,47$
Class number $1$
Class group trivial
Galois group $D_5\times S_3$ (as 15T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1)
 
gp: K = bnfinit(y^15 - 4*y^14 + 3*y^13 + 15*y^12 - 40*y^11 + 23*y^10 + 52*y^9 - 111*y^8 + 83*y^7 - 67*y^5 + 86*y^4 - 62*y^3 + 26*y^2 - 7*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1)
 

\( x^{15} - 4 x^{14} + 3 x^{13} + 15 x^{12} - 40 x^{11} + 23 x^{10} + 52 x^{9} - 111 x^{8} + 83 x^{7} + \cdots + 1 \) 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:   \(126064044311049216\) \(\medspace = 2^{10}\cdot 3^{5}\cdot 47^{7}\) 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:  \(13.81\)
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:  $2^{2/3}3^{1/2}47^{1/2}\approx 18.849343120394256$
Ramified primes:   \(2\), \(3\), \(47\) 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:  \(\Q(\sqrt{141}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{404281}a^{14}+\frac{182213}{404281}a^{13}-\frac{79463}{404281}a^{12}-\frac{185441}{404281}a^{11}+\frac{111805}{404281}a^{10}-\frac{160725}{404281}a^{9}+\frac{96929}{404281}a^{8}-\frac{116846}{404281}a^{7}+\frac{131366}{404281}a^{6}+\frac{44693}{404281}a^{5}-\frac{12150}{404281}a^{4}-\frac{93708}{404281}a^{3}+\frac{21618}{404281}a^{2}-\frac{1348}{3709}a+\frac{974}{404281}$ 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:   $\frac{576707}{404281}a^{14}-\frac{2056301}{404281}a^{13}+\frac{808695}{404281}a^{12}+\frac{9032887}{404281}a^{11}-\frac{19051762}{404281}a^{10}+\frac{4740791}{404281}a^{9}+\frac{32041413}{404281}a^{8}-\frac{49671324}{404281}a^{7}+\frac{26136313}{404281}a^{6}+\frac{11150664}{404281}a^{5}-\frac{33951362}{404281}a^{4}+\frac{35375566}{404281}a^{3}-\frac{20583883}{404281}a^{2}+\frac{53581}{3709}a-\frac{1046534}{404281}$, $\frac{188233}{404281}a^{14}-\frac{700411}{404281}a^{13}+\frac{433840}{404281}a^{12}+\frac{2740035}{404281}a^{11}-\frac{6733948}{404281}a^{10}+\frac{3449677}{404281}a^{9}+\frac{8929109}{404281}a^{8}-\frac{18366520}{404281}a^{7}+\frac{14123029}{404281}a^{6}-\frac{794422}{404281}a^{5}-\frac{11333201}{404281}a^{4}+\frac{15204744}{404281}a^{3}-\frac{11187139}{404281}a^{2}+\frac{42823}{3709}a-\frac{1013194}{404281}$, $a$, $a-1$, $\frac{409864}{404281}a^{14}-\frac{1492941}{404281}a^{13}+\frac{662890}{404281}a^{12}+\frac{6515034}{404281}a^{11}-\frac{14152384}{404281}a^{10}+\frac{3814674}{404281}a^{9}+\frac{24079208}{404281}a^{8}-\frac{37439817}{404281}a^{7}+\frac{18243289}{404281}a^{6}+\frac{10590948}{404281}a^{5}-\frac{25383945}{404281}a^{4}+\frac{24628991}{404281}a^{3}-\frac{13932279}{404281}a^{2}+\frac{29349}{3709}a-\frac{222092}{404281}$, $\frac{228992}{404281}a^{14}-\frac{926995}{404281}a^{13}+\frac{700795}{404281}a^{12}+\frac{3596454}{404281}a^{11}-\frac{9559352}{404281}a^{10}+\frac{5045850}{404281}a^{9}+\frac{13875660}{404281}a^{8}-\frac{27356636}{404281}a^{7}+\frac{17002226}{404281}a^{6}+\frac{5221594}{404281}a^{5}-\frac{18587884}{404281}a^{4}+\frac{18641508}{404281}a^{3}-\frac{11391657}{404281}a^{2}+\frac{29981}{3709}a-\frac{124904}{404281}$, $\frac{149248}{404281}a^{14}-\frac{652765}{404281}a^{13}+\frac{697873}{404281}a^{12}+\frac{1996016}{404281}a^{11}-\frac{6898412}{404281}a^{10}+\frac{6192550}{404281}a^{9}+\frac{6136584}{404281}a^{8}-\frac{20584923}{404281}a^{7}+\frac{20315442}{404281}a^{6}-\frac{4338446}{404281}a^{5}-\frac{11887064}{404281}a^{4}+\frac{17753694}{404281}a^{3}-\frac{14273232}{404281}a^{2}+\frac{60327}{3709}a-\frac{1386451}{404281}$, $\frac{114322}{404281}a^{14}-\frac{432501}{404281}a^{13}+\frac{229265}{404281}a^{12}+\frac{1722481}{404281}a^{11}-\frac{4019696}{404281}a^{10}+\frac{1785124}{404281}a^{9}+\frac{5434862}{404281}a^{8}-\frac{10731197}{404281}a^{7}+\frac{8687446}{404281}a^{6}-\frac{718694}{404281}a^{5}-\frac{7584123}{404281}a^{4}+\frac{10263268}{404281}a^{3}-\frac{6425253}{404281}a^{2}+\frac{25148}{3709}a-\frac{636209}{404281}$ 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:  \( 280.863045663 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 280.863045663 \cdot 1}{2\cdot\sqrt{126064044311049216}}\cr\approx \mathstrut & 0.194687552111 \end{aligned}\]

# 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 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 4*x^14 + 3*x^13 + 15*x^12 - 40*x^11 + 23*x^10 + 52*x^9 - 111*x^8 + 83*x^7 - 67*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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

$S_3\times D_5$ (as 15T7):

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 60
The 12 conjugacy class representatives for $D_5\times S_3$
Character table for $D_5\times S_3$

Intermediate fields

3.3.564.1, 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

Degree 30 siblings: 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 ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ $15$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $15$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ R ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.15.10.1$x^{15} + 13 x^{12} + 3 x^{10} + 13 x^{9} - 174 x^{7} - 45 x^{6} + 3 x^{5} + 513 x^{4} + 63 x^{2} - 216 x + 109$$3$$5$$10$$S_3 \times C_5$$[\ ]_{3}^{10}$
\(3\) Copy content Toggle raw display 3.5.0.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} + 235$$2$$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$$[\ ]_{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$