↑

Properties

Label 15.1.24118280788986467.1
Degree $15$
Signature $[1, 7]$
Discriminant $-2.412\times 10^{16}$
Root discriminant \(12.36\)
Ramified primes $59,83$
Class number $1$
Class group trivial
Galois group $S_5 \times S_3$ (as 15T29)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + x + 1)
 
gp: K = bnfinit(y^15 - 2*y^14 + 6*y^12 - 13*y^11 + 13*y^10 - y^9 - 21*y^8 + 40*y^7 - 41*y^6 + 29*y^5 - 18*y^4 + 13*y^3 - 7*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + x + 1)
 

\( x^{15} - 2 x^{14} + 6 x^{12} - 13 x^{11} + 13 x^{10} - x^{9} - 21 x^{8} + 40 x^{7} - 41 x^{6} + 29 x^{5} + \cdots + 1 \) 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:  $[1, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-24118280788986467\) \(\medspace = -\,59^{6}\cdot 83^{3}\) 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:  \(12.36\)
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:  $59^{1/2}83^{1/2}\approx 69.97856814768362$
Ramified primes:   \(59\), \(83\) 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:  \(\Q(\sqrt{-83}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{28613}a^{14}-\frac{3491}{28613}a^{13}-\frac{9039}{28613}a^{12}+\frac{427}{2201}a^{11}+\frac{273}{2201}a^{10}+\frac{537}{2201}a^{9}-\frac{7047}{28613}a^{8}+\frac{8395}{28613}a^{7}+\frac{9597}{28613}a^{6}-\frac{6764}{28613}a^{5}-\frac{6100}{28613}a^{4}-\frac{5190}{28613}a^{3}-\frac{4106}{28613}a^{2}-\frac{9286}{28613}a+\frac{8939}{28613}$ 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:  $7$
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:   $\frac{5066}{28613}a^{14}-\frac{2572}{28613}a^{13}-\frac{10774}{28613}a^{12}+\frac{1800}{2201}a^{11}-\frac{1411}{2201}a^{10}+\frac{6}{2201}a^{9}+\frac{37535}{28613}a^{8}-\frac{75687}{28613}a^{7}+\frac{62141}{28613}a^{6}-\frac{16663}{28613}a^{5}-\frac{560}{28613}a^{4}+\frac{2807}{28613}a^{3}-\frac{27958}{28613}a^{2}+\frac{54122}{28613}a-\frac{9405}{28613}$, $\frac{237}{403}a^{14}-\frac{411}{403}a^{13}-\frac{298}{403}a^{12}+\frac{108}{31}a^{11}-\frac{182}{31}a^{10}+\frac{138}{31}a^{9}+\frac{1102}{403}a^{8}-\frac{4832}{403}a^{7}+\frac{7211}{403}a^{6}-\frac{5576}{403}a^{5}+\frac{3085}{403}a^{4}-\frac{2089}{403}a^{3}+\frac{1332}{403}a^{2}-\frac{402}{403}a-\frac{28}{403}$, $\frac{28}{403}a^{14}+\frac{181}{403}a^{13}-\frac{411}{403}a^{12}-\frac{10}{31}a^{11}+\frac{80}{31}a^{10}-\frac{154}{31}a^{9}+\frac{1766}{403}a^{8}+\frac{514}{403}a^{7}-\frac{3712}{403}a^{6}+\frac{6063}{403}a^{5}-\frac{4764}{403}a^{4}+\frac{2581}{403}a^{3}-\frac{1725}{403}a^{2}+\frac{1136}{403}a-\frac{374}{403}$, $\frac{4962}{28613}a^{14}+\frac{17136}{28613}a^{13}-\frac{14947}{28613}a^{12}-\frac{789}{2201}a^{11}+\frac{3212}{2201}a^{10}-\frac{7420}{2201}a^{9}+\frac{55098}{28613}a^{8}+\frac{24075}{28613}a^{7}-\frac{163396}{28613}a^{6}+\frac{228985}{28613}a^{5}-\frac{138711}{28613}a^{4}+\frac{141985}{28613}a^{3}-\frac{87355}{28613}a^{2}+\frac{75637}{28613}a+\frac{5168}{28613}$, $\frac{14906}{28613}a^{14}-\frac{18412}{28613}a^{13}-\frac{25330}{28613}a^{12}+\frac{6173}{2201}a^{11}-\frac{9115}{2201}a^{10}+\frac{3887}{2201}a^{9}+\frac{110193}{28613}a^{8}-\frac{274909}{28613}a^{7}+\frac{331238}{28613}a^{6}-\frac{163650}{28613}a^{5}+\frac{62740}{28613}a^{4}-\frac{49814}{28613}a^{3}+\frac{56397}{28613}a^{2}-\frac{16035}{28613}a-\frac{34620}{28613}$, $\frac{5238}{28613}a^{14}-\frac{2151}{28613}a^{13}-\frac{20380}{28613}a^{12}+\frac{2611}{2201}a^{11}-\frac{676}{2201}a^{10}-\frac{4474}{2201}a^{9}+\frac{113036}{28613}a^{8}-\frac{119623}{28613}a^{7}-\frac{3955}{28613}a^{6}+\frac{164740}{28613}a^{5}-\frac{219983}{28613}a^{4}+\frac{140195}{28613}a^{3}-\frac{76091}{28613}a^{2}+\frac{87871}{28613}a-\frac{45612}{28613}$, $\frac{24996}{28613}a^{14}-\frac{19999}{28613}a^{13}-\frac{39209}{28613}a^{12}+\frac{9447}{2201}a^{11}-\frac{12398}{2201}a^{10}+\frac{3355}{2201}a^{9}+\frac{166494}{28613}a^{8}-\frac{406904}{28613}a^{7}+\frac{424415}{28613}a^{6}-\frac{199018}{28613}a^{5}+\frac{88916}{28613}a^{4}-\frac{83737}{28613}a^{3}+\frac{115707}{28613}a^{2}+\frac{24413}{28613}a-\frac{28286}{28613}$ 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:  \( 81.1679989991 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 81.1679989991 \cdot 1}{2\cdot\sqrt{24118280788986467}}\cr\approx \mathstrut & 0.202055531680 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + 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 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + 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 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + 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 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + 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

3.1.59.1, 5.1.4897.1

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 ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ $15$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.3.0.1}{3} }^{5}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.3.0.1}{3} }^{5}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ R

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
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(83\) Copy content Toggle raw display 83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.3.0.1$x^{3} + 3 x + 81$$1$$3$$0$$C_3$$[\ ]^{3}$
83.4.2.1$x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.6.0.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$