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Properties

Label 15.5.77595502265817783.1
Degree $15$
Signature $[5, 5]$
Discriminant $-7.760\times 10^{16}$
Root discriminant \(13.37\)
Ramified primes $3,13,11772948303113$
Class number $1$
Class group trivial
Galois group $S_{15}$ (as 15T104)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^15 - 3*y^13 - 2*y^12 + y^11 + 12*y^10 + 3*y^9 - 15*y^8 - 10*y^7 - 4*y^6 + 20*y^5 + 7*y^4 - 3*y^3 - 7*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*x + 1)
 

\( x^{15} - 3 x^{13} - 2 x^{12} + x^{11} + 12 x^{10} + 3 x^{9} - 15 x^{8} - 10 x^{7} - 4 x^{6} + 20 x^{5} + \cdots + 1 \) 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:  $[5, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-77595502265817783\) \(\medspace = -\,3\cdot 13^{3}\cdot 11772948303113\) 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:  \(13.37\)
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:  $3^{1/2}13^{3/4}11772948303113^{1/2}\approx 40687476.96822902$
Ramified primes:   \(3\), \(13\), \(11772948303113\) 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:  $\Q(\sqrt{-459144983821407}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{136709}a^{14}+\frac{29762}{136709}a^{13}+\frac{39030}{136709}a^{12}-\frac{5515}{136709}a^{11}+\frac{50080}{136709}a^{10}-\frac{57255}{136709}a^{9}+\frac{54378}{136709}a^{8}+\frac{36879}{136709}a^{7}-\frac{43773}{136709}a^{6}+\frac{64740}{136709}a^{5}+\frac{15254}{136709}a^{4}-\frac{21034}{136709}a^{3}-\frac{23400}{136709}a^{2}-\frac{35161}{136709}a+\frac{45709}{136709}$ 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:  $9$
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:   $\frac{23519}{136709}a^{14}+\frac{22398}{136709}a^{13}-\frac{54365}{136709}a^{12}-\frac{107153}{136709}a^{11}-\frac{53224}{136709}a^{10}+\frac{276723}{136709}a^{9}+\frac{276905}{136709}a^{8}-\frac{198113}{136709}a^{7}-\frac{488544}{136709}a^{6}-\frac{318200}{136709}a^{5}+\frac{444537}{136709}a^{4}+\frac{324643}{136709}a^{3}+\frac{45834}{136709}a^{2}-\frac{272236}{136709}a-\frac{49605}{136709}$, $\frac{29105}{136709}a^{14}+\frac{34786}{136709}a^{13}-\frac{83640}{136709}a^{12}-\frac{154418}{136709}a^{11}-\frac{12958}{136709}a^{10}+\frac{349353}{136709}a^{9}+\frac{401724}{136709}a^{8}-\frac{349191}{136709}a^{7}-\frac{705539}{136709}a^{6}-\frac{139156}{136709}a^{5}+\frac{346965}{136709}a^{4}+\frac{535168}{136709}a^{3}+\frac{27238}{136709}a^{2}-\frac{367458}{136709}a-\frac{91543}{136709}$, $\frac{7345}{136709}a^{14}+\frac{4199}{136709}a^{13}-\frac{3423}{136709}a^{12}-\frac{41811}{136709}a^{11}-\frac{46319}{136709}a^{10}+\frac{115618}{136709}a^{9}+\frac{79421}{136709}a^{8}+\frac{55726}{136709}a^{7}-\frac{246535}{136709}a^{6}-\frac{232020}{136709}a^{5}+\frac{349377}{136709}a^{4}-\frac{13560}{136709}a^{3}+\frac{243631}{136709}a^{2}-\frac{287662}{136709}a-\frac{24699}{136709}$, $\frac{40610}{136709}a^{14}-\frac{9449}{136709}a^{13}-\frac{132555}{136709}a^{12}-\frac{34808}{136709}a^{11}+\frac{65716}{136709}a^{10}+\frac{431249}{136709}a^{9}+\frac{30103}{136709}a^{8}-\frac{674450}{136709}a^{7}-\frac{131112}{136709}a^{6}-\frac{96088}{136709}a^{5}+\frac{583297}{136709}a^{4}+\frac{240510}{136709}a^{3}-\frac{146450}{136709}a^{2}-\frac{99414}{136709}a-\frac{129021}{136709}$, $\frac{3675}{136709}a^{14}+\frac{8150}{136709}a^{13}+\frac{27509}{136709}a^{12}-\frac{34693}{136709}a^{11}-\frac{103023}{136709}a^{10}-\frac{16974}{136709}a^{9}+\frac{107301}{136709}a^{8}+\frac{325124}{136709}a^{7}-\frac{95991}{136709}a^{6}-\frac{364287}{136709}a^{5}-\frac{128949}{136709}a^{4}-\frac{59365}{136709}a^{3}+\frac{268379}{136709}a^{2}-\frac{26670}{136709}a+\frac{101923}{136709}$, $\frac{41850}{136709}a^{14}-\frac{15999}{136709}a^{13}-\frac{130341}{136709}a^{12}-\frac{37958}{136709}a^{11}+\frac{99030}{136709}a^{10}+\frac{523729}{136709}a^{9}-\frac{75423}{136709}a^{8}-\frac{742005}{136709}a^{7}-\frac{272868}{136709}a^{6}+\frac{70038}{136709}a^{5}+\frac{1042542}{136709}a^{4}-\frac{3649}{136709}a^{3}-\frac{180142}{136709}a^{2}-\frac{499010}{136709}a-\frac{47387}{136709}$, $a$, $\frac{1475}{136709}a^{14}+\frac{15361}{136709}a^{13}+\frac{14761}{136709}a^{12}-\frac{68794}{136709}a^{11}-\frac{91569}{136709}a^{10}+\frac{35037}{136709}a^{9}+\frac{232785}{136709}a^{8}+\frac{259761}{136709}a^{7}-\frac{311945}{136709}a^{6}-\frac{478218}{136709}a^{5}-\frac{57335}{136709}a^{4}+\frac{281211}{136709}a^{3}+\frac{482504}{136709}a^{2}-\frac{49764}{136709}a-\frac{113471}{136709}$, $\frac{4199}{136709}a^{14}+\frac{18612}{136709}a^{13}-\frac{27121}{136709}a^{12}-\frac{53664}{136709}a^{11}+\frac{27478}{136709}a^{10}+\frac{57386}{136709}a^{9}+\frac{165901}{136709}a^{8}-\frac{173085}{136709}a^{7}-\frac{202640}{136709}a^{6}+\frac{202477}{136709}a^{5}-\frac{64975}{136709}a^{4}+\frac{128957}{136709}a^{3}-\frac{236247}{136709}a^{2}+\frac{4681}{136709}a+\frac{129364}{136709}$ 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:  \( 416.964032767 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 416.964032767 \cdot 1}{2\cdot\sqrt{77595502265817783}}\cr\approx \mathstrut & 0.234530670183 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*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 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*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 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*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 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*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

$S_{15}$ (as 15T104):

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 1307674368000
The 176 conjugacy class representatives for $S_{15}$
Character table for $S_{15}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 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 $15$ R ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.5.0.1}{5} }^{3}$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $15$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.7.0.1}{7} }$

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
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.5.0.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.8.0.1$x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
\(13\) Copy content Toggle raw display 13.4.0.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.4$x^{4} + 91$$4$$1$$3$$C_4$$[\ ]_{4}$
13.7.0.1$x^{7} + 3 x + 11$$1$$7$$0$$C_7$$[\ ]^{7}$
\(11772948303113\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$