Properties

Label 15.3.124...288.1
Degree $15$
Signature $[3, 6]$
Discriminant $1.241\times 10^{18}$
Root discriminant \(16.08\)
Ramified primes $2,13$
Class number $1$
Class group trivial
Galois group $F_5\times C_3$ (as 15T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*x^2 + 6*x - 1)
 
gp: K = bnfinit(y^15 - 5*y^14 + 16*y^13 - 39*y^12 + 65*y^11 - 91*y^10 + 78*y^9 - 39*y^8 - 26*y^7 + 78*y^6 - 78*y^5 + 52*y^4 - 13*y^3 - 9*y^2 + 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*x^2 + 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*x^2 + 6*x - 1)
 

\( x^{15} - 5 x^{14} + 16 x^{13} - 39 x^{12} + 65 x^{11} - 91 x^{10} + 78 x^{9} - 39 x^{8} - 26 x^{7} + 78 x^{6} - 78 x^{5} + 52 x^{4} - 13 x^{3} - 9 x^{2} + 6 x - 1 \) 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:   \(1240576436601868288\) \(\medspace = 2^{12}\cdot 13^{13}\) 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:  \(16.08\)
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:  $2^{4/5}13^{11/12}\approx 18.278423529398573$
Ramified primes:   \(2\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}-\frac{4}{13}a^{11}+\frac{3}{13}a^{10}+\frac{1}{13}a^{9}-\frac{4}{13}a^{8}+\frac{3}{13}a^{7}+\frac{1}{13}a^{6}-\frac{4}{13}a^{5}+\frac{3}{13}a^{4}+\frac{1}{13}a^{3}-\frac{4}{13}a^{2}+\frac{3}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{4}{13}$, $\frac{1}{403}a^{14}+\frac{6}{403}a^{13}-\frac{11}{403}a^{12}+\frac{57}{403}a^{11}-\frac{176}{403}a^{10}-\frac{167}{403}a^{9}+\frac{70}{403}a^{8}-\frac{137}{403}a^{7}-\frac{76}{403}a^{6}-\frac{138}{403}a^{5}-\frac{46}{403}a^{4}+\frac{197}{403}a^{3}-\frac{47}{403}a^{2}-\frac{185}{403}a-\frac{13}{31}$ 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:   $\frac{3}{13}a^{14}-\frac{4}{13}a^{13}+\frac{10}{13}a^{12}-\frac{14}{13}a^{11}-\frac{22}{13}a^{10}-\frac{29}{13}a^{9}-\frac{118}{13}a^{8}-\frac{74}{13}a^{7}-\frac{146}{13}a^{6}-\frac{40}{13}a^{5}+\frac{4}{13}a^{4}+\frac{10}{13}a^{3}+\frac{103}{13}a^{2}-\frac{10}{13}a-\frac{6}{13}$, $\frac{72}{403}a^{14}-\frac{219}{403}a^{13}+\frac{44}{31}a^{12}-\frac{73}{31}a^{11}+\frac{22}{31}a^{10}+\frac{17}{31}a^{9}-\frac{342}{31}a^{8}+\frac{269}{31}a^{7}-\frac{471}{31}a^{6}-\frac{37}{31}a^{5}+\frac{184}{31}a^{4}-\frac{292}{31}a^{3}+\frac{281}{31}a^{2}+\frac{444}{403}a-\frac{512}{403}$, $\frac{72}{403}a^{14}-\frac{219}{403}a^{13}+\frac{44}{31}a^{12}-\frac{73}{31}a^{11}+\frac{22}{31}a^{10}+\frac{17}{31}a^{9}-\frac{342}{31}a^{8}+\frac{269}{31}a^{7}-\frac{471}{31}a^{6}-\frac{37}{31}a^{5}+\frac{184}{31}a^{4}-\frac{292}{31}a^{3}+\frac{250}{31}a^{2}+\frac{847}{403}a-\frac{512}{403}$, $\frac{96}{403}a^{14}-\frac{32}{31}a^{13}+\frac{1145}{403}a^{12}-\frac{2526}{403}a^{11}+\frac{3409}{403}a^{10}-\frac{4159}{403}a^{9}+\frac{2349}{403}a^{8}-\frac{101}{403}a^{7}-\frac{1468}{403}a^{6}+\frac{3337}{403}a^{5}-\frac{2246}{403}a^{4}+\frac{1366}{403}a^{3}-\frac{1629}{403}a^{2}+\frac{127}{403}a+\frac{144}{403}$, $\frac{209}{403}a^{14}-\frac{978}{403}a^{13}+\frac{3002}{403}a^{12}-\frac{7276}{403}a^{11}+\frac{11762}{403}a^{10}-\frac{16706}{403}a^{9}+\frac{14382}{403}a^{8}-\frac{7088}{403}a^{7}-\frac{3732}{403}a^{6}+\frac{16449}{403}a^{5}-\frac{14667}{403}a^{4}+\frac{10607}{403}a^{3}-\frac{2414}{403}a^{2}-\frac{3821}{403}a+\frac{104}{31}$, $\frac{73}{403}a^{14}-\frac{492}{403}a^{13}+\frac{1584}{403}a^{12}-\frac{3775}{403}a^{11}+\frac{6403}{403}a^{10}-\frac{7386}{403}a^{9}+\frac{6040}{403}a^{8}+\frac{384}{403}a^{7}-\frac{4773}{403}a^{6}+\frac{6976}{403}a^{5}-\frac{6272}{403}a^{4}+\frac{245}{403}a^{3}-\frac{486}{403}a^{2}-\frac{54}{31}a+\frac{435}{403}$, $\frac{491}{403}a^{14}-\frac{2200}{403}a^{13}+\frac{6720}{403}a^{12}-\frac{15661}{403}a^{11}+\frac{23696}{403}a^{10}-\frac{31994}{403}a^{9}+\frac{20544}{403}a^{8}-\frac{6321}{403}a^{7}-\frac{19150}{403}a^{6}+\frac{31256}{403}a^{5}-\frac{23299}{403}a^{4}+\frac{12531}{403}a^{3}+\frac{2591}{403}a^{2}-\frac{5306}{403}a+\frac{96}{31}$, $\frac{824}{403}a^{14}-\frac{3612}{403}a^{13}+\frac{11117}{403}a^{12}-\frac{25696}{403}a^{11}+\frac{38434}{403}a^{10}-\frac{52141}{403}a^{9}+\frac{30555}{403}a^{8}-\frac{10836}{403}a^{7}-\frac{35189}{403}a^{6}+\frac{45752}{403}a^{5}-\frac{36199}{403}a^{4}+\frac{17279}{403}a^{3}+\frac{5478}{403}a^{2}-\frac{404}{31}a+\frac{1050}{403}$ 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:  \( 1735.38854006 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 1735.38854006 \cdot 1}{2\cdot\sqrt{1240576436601868288}}\cr\approx \mathstrut & 0.383463615476 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*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 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*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 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*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 - 5*x^14 + 16*x^13 - 39*x^12 + 65*x^11 - 91*x^10 + 78*x^9 - 39*x^8 - 26*x^7 + 78*x^6 - 78*x^5 + 52*x^4 - 13*x^3 - 9*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

$C_3\times F_5$ (as 15T8):

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 15 conjugacy class representatives for $F_5\times C_3$
Character table for $F_5\times C_3$

Intermediate fields

3.3.169.1, 5.1.35152.1

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 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 $15$ ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ R ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }$ $15$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.5.0.1}{5} }^{3}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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}$
\(13\) Copy content Toggle raw display 13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.12.11.4$x^{12} + 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$