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Properties

Label 15.3.15315650494266781.1
Degree $15$
Signature $[3, 6]$
Discriminant $1.532\times 10^{16}$
Root discriminant \(12.00\)
Ramified primes $127,9829$
Class number $1$
Class group trivial
Galois group $C_3\wr S_5$ (as 15T78)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*x - 1)
 
gp: K = bnfinit(y^15 - 2*y^14 + y^13 - y^12 + 7*y^11 - 5*y^10 - 10*y^9 + 7*y^8 + 16*y^7 - 15*y^6 - 19*y^5 + 27*y^4 - 2*y^3 - 11*y^2 + 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*x - 1)
 

\( x^{15} - 2 x^{14} + x^{13} - x^{12} + 7 x^{11} - 5 x^{10} - 10 x^{9} + 7 x^{8} + 16 x^{7} - 15 x^{6} + \cdots - 1 \) 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:   \(15315650494266781\) \(\medspace = 127^{2}\cdot 9829^{3}\) 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:  \(12.00\)
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:  $127^{2/3}9829^{1/2}\approx 2504.9005119012204$
Ramified primes:   \(127\), \(9829\) 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:  \(\Q(\sqrt{9829}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2962}a^{14}+\frac{60}{1481}a^{13}-\frac{169}{2962}a^{12}-\frac{683}{1481}a^{11}+\frac{354}{1481}a^{10}+\frac{473}{2962}a^{9}+\frac{709}{1481}a^{8}+\frac{1207}{2962}a^{7}+\frac{651}{2962}a^{6}+\frac{457}{1481}a^{5}-\frac{1067}{2962}a^{4}-\frac{650}{1481}a^{3}-\frac{135}{2962}a^{2}+\frac{1291}{2962}a+\frac{261}{1481}$ 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:   $\frac{8454}{1481}a^{14}-\frac{14815}{1481}a^{13}+\frac{3401}{1481}a^{12}-\frac{6731}{1481}a^{11}+\frac{58470}{1481}a^{10}-\frac{26616}{1481}a^{9}-\frac{98669}{1481}a^{8}+\frac{29508}{1481}a^{7}+\frac{157144}{1481}a^{6}-\frac{79395}{1481}a^{5}-\frac{199582}{1481}a^{4}+\frac{170616}{1481}a^{3}+\frac{56839}{1481}a^{2}-\frac{77868}{1481}a+\frac{17380}{1481}$, $a$, $\frac{11329}{2962}a^{14}-\frac{7483}{2962}a^{13}-\frac{7073}{2962}a^{12}-\frac{12293}{2962}a^{11}+\frac{32500}{1481}a^{10}+\frac{38865}{2962}a^{9}-\frac{112441}{2962}a^{8}-\frac{81425}{2962}a^{7}+\frac{79133}{1481}a^{6}+\frac{72123}{2962}a^{5}-\frac{245967}{2962}a^{4}-\frac{31737}{2962}a^{3}+\frac{120419}{2962}a^{2}-\frac{4011}{1481}a-\frac{14705}{2962}$, $\frac{10377}{1481}a^{14}-\frac{12129}{1481}a^{13}-\frac{1899}{2962}a^{12}-\frac{19915}{2962}a^{11}+\frac{64839}{1481}a^{10}+\frac{3249}{1481}a^{9}-\frac{108743}{1481}a^{8}-\frac{39543}{2962}a^{7}+\frac{325511}{2962}a^{6}-\frac{14556}{1481}a^{5}-\frac{458235}{2962}a^{4}+\frac{173935}{2962}a^{3}+\frac{78624}{1481}a^{2}-\frac{52254}{1481}a+\frac{14883}{2962}$, $\frac{3602}{1481}a^{14}-\frac{7829}{2962}a^{13}-\frac{47}{1481}a^{12}-\frac{8305}{2962}a^{11}+\frac{22149}{1481}a^{10}+\frac{2077}{1481}a^{9}-\frac{67311}{2962}a^{8}-\frac{9488}{1481}a^{7}+\frac{100185}{2962}a^{6}-\frac{7475}{2962}a^{5}-\frac{71227}{1481}a^{4}+\frac{49517}{2962}a^{3}+\frac{18751}{1481}a^{2}-\frac{28455}{2962}a+\frac{9115}{2962}$, $\frac{3160}{1481}a^{14}-\frac{16163}{2962}a^{13}+\frac{8607}{2962}a^{12}-\frac{2407}{1481}a^{11}+\frac{24666}{1481}a^{10}-\frac{26307}{1481}a^{9}-\frac{73821}{2962}a^{8}+\frac{73659}{2962}a^{7}+\frac{65215}{1481}a^{6}-\frac{140115}{2962}a^{5}-\frac{142623}{2962}a^{4}+\frac{118774}{1481}a^{3}+\frac{2890}{1481}a^{2}-\frac{97455}{2962}a+\frac{14496}{1481}$, $\frac{14803}{1481}a^{14}-\frac{44629}{2962}a^{13}+\frac{6809}{2962}a^{12}-\frac{12653}{1481}a^{11}+\frac{97233}{1481}a^{10}-\frac{25526}{1481}a^{9}-\frac{326419}{2962}a^{8}+\frac{46785}{2962}a^{7}+\frac{251656}{1481}a^{6}-\frac{194959}{2962}a^{5}-\frac{667803}{2962}a^{4}+\frac{235693}{1481}a^{3}+\frac{94248}{1481}a^{2}-\frac{238743}{2962}a+\frac{28928}{1481}$, $\frac{39037}{2962}a^{14}-\frac{53279}{2962}a^{13}+\frac{5045}{2962}a^{12}-\frac{36681}{2962}a^{11}+\frac{125772}{1481}a^{10}-\frac{36151}{2962}a^{9}-\frac{412589}{2962}a^{8}+\frac{4087}{2962}a^{7}+\frac{315757}{1481}a^{6}-\frac{176673}{2962}a^{5}-\frac{856853}{2962}a^{4}+\frac{495981}{2962}a^{3}+\frac{248211}{2962}a^{2}-\frac{128938}{1481}a+\frac{56513}{2962}$ 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:  \( 87.5727798613 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 87.5727798613 \cdot 1}{2\cdot\sqrt{15315650494266781}}\cr\approx \mathstrut & 0.174156871209 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*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 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*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 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*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 - 2*x^14 + x^13 - x^12 + 7*x^11 - 5*x^10 - 10*x^9 + 7*x^8 + 16*x^7 - 15*x^6 - 19*x^5 + 27*x^4 - 2*x^3 - 11*x^2 + 6*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

$C_3\wr S_5$ (as 15T78):

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 29160
The 108 conjugacy class representatives for $C_3\wr S_5$
Character table for $C_3\wr S_5$

Intermediate fields

5.1.9829.1

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 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ $15$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ $15$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ $15$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.5.0.1}{5} }^{3}$ ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ $15$ ${\href{/padicField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
\(127\) Copy content Toggle raw display 127.3.2.1$x^{3} + 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.0.1$x^{3} + 3 x + 124$$1$$3$$0$$C_3$$[\ ]^{3}$
127.9.0.1$x^{9} + 14 x^{3} + 119 x^{2} + 126 x + 124$$1$$9$$0$$C_9$$[\ ]^{9}$
\(9829\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$