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Properties

Label 15.1.123...896.1
Degree $15$
Signature $[1, 7]$
Discriminant $-1.233\times 10^{23}$
Root discriminant \(34.62\)
Ramified primes $2,739$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76)
 
gp: K = bnfinit(y^15 + y^13 - 18*y^12 + 32*y^11 - 50*y^10 + 129*y^9 - 114*y^8 + 232*y^7 - 146*y^6 + 73*y^5 + 238*y^4 - 249*y^3 + 270*y^2 - 232*y + 76, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76)
 

\( x^{15} + x^{13} - 18 x^{12} + 32 x^{11} - 50 x^{10} + 129 x^{9} - 114 x^{8} + 232 x^{7} - 146 x^{6} + \cdots + 76 \) 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:   \(-123256838893351061072896\) \(\medspace = -\,2^{10}\cdot 739^{7}\) 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:  \(34.62\)
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:  $2^{2/3}739^{1/2}\approx 43.15279031238447$
Ramified primes:   \(2\), \(739\) 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:  \(\Q(\sqrt{-739}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}-\frac{2}{9}a^{6}+\frac{7}{18}a^{5}-\frac{2}{9}a^{4}+\frac{1}{18}a^{3}-\frac{1}{18}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{18}a^{9}+\frac{1}{18}a^{7}+\frac{5}{18}a^{6}-\frac{5}{18}a^{5}-\frac{1}{18}a^{4}+\frac{2}{9}a^{3}-\frac{1}{18}a^{2}+\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{18}a^{10}-\frac{1}{9}a^{7}+\frac{5}{18}a^{6}+\frac{2}{9}a^{5}-\frac{2}{9}a^{4}-\frac{4}{9}a^{3}-\frac{5}{18}a^{2}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{2}a^{3}+\frac{1}{9}a^{2}+\frac{2}{9}a-\frac{2}{9}$, $\frac{1}{18}a^{12}+\frac{1}{18}a^{7}+\frac{4}{9}a^{6}+\frac{1}{18}a^{5}-\frac{5}{18}a^{4}+\frac{1}{18}a^{3}+\frac{5}{18}a^{2}-\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{108}a^{13}-\frac{1}{108}a^{12}+\frac{1}{108}a^{11}+\frac{1}{108}a^{9}-\frac{1}{36}a^{8}+\frac{5}{108}a^{7}-\frac{5}{27}a^{6}+\frac{37}{108}a^{5}-\frac{25}{108}a^{4}-\frac{23}{108}a^{3}+\frac{5}{54}a^{2}+\frac{13}{27}a-\frac{2}{27}$, $\frac{1}{7382528244}a^{14}-\frac{5207093}{7382528244}a^{13}+\frac{3618689}{567886788}a^{12}-\frac{12085708}{1845632061}a^{11}+\frac{70377847}{7382528244}a^{10}+\frac{196018301}{7382528244}a^{9}+\frac{69211157}{7382528244}a^{8}-\frac{6852388}{141971697}a^{7}+\frac{215716301}{2460842748}a^{6}-\frac{37131173}{567886788}a^{5}+\frac{900310289}{7382528244}a^{4}-\frac{59551439}{410140458}a^{3}-\frac{382248367}{1230421374}a^{2}+\frac{293311855}{615210687}a+\frac{363309587}{1845632061}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:   $\frac{16179710}{1845632061}a^{14}+\frac{20924012}{1845632061}a^{13}+\frac{428950}{141971697}a^{12}-\frac{642344323}{3691264122}a^{11}+\frac{104456528}{1845632061}a^{10}+\frac{134397311}{3691264122}a^{9}+\frac{3302405591}{3691264122}a^{8}+\frac{29490569}{283943394}a^{7}+\frac{114577451}{410140458}a^{6}-\frac{24087940}{141971697}a^{5}-\frac{2667600907}{3691264122}a^{4}+\frac{423602588}{615210687}a^{3}-\frac{202571006}{615210687}a^{2}+\frac{364252870}{615210687}a-\frac{28042817}{1845632061}$, $\frac{693055}{136713486}a^{14}-\frac{11153}{2460842748}a^{13}+\frac{2270987}{189295596}a^{12}-\frac{233861135}{2460842748}a^{11}+\frac{69064559}{410140458}a^{10}-\frac{949180937}{2460842748}a^{9}+\frac{86357625}{91142324}a^{8}-\frac{195657601}{189295596}a^{7}+\frac{2839410059}{1230421374}a^{6}-\frac{420536675}{189295596}a^{5}+\frac{6942915719}{2460842748}a^{4}-\frac{4424298761}{2460842748}a^{3}+\frac{945086929}{615210687}a^{2}+\frac{83921776}{615210687}a-\frac{143244992}{615210687}$, $\frac{20412890}{1845632061}a^{14}+\frac{27149305}{3691264122}a^{13}+\frac{261055}{141971697}a^{12}-\frac{376922384}{1845632061}a^{11}+\frac{748136965}{3691264122}a^{10}-\frac{635032609}{3691264122}a^{9}+\frac{1818365557}{1845632061}a^{8}-\frac{9886487}{141971697}a^{7}+\frac{1263677869}{1230421374}a^{6}-\frac{84858827}{283943394}a^{5}-\frac{3908559896}{1845632061}a^{4}+\frac{1098268468}{615210687}a^{3}-\frac{440897243}{410140458}a^{2}-\frac{74293097}{205070229}a+\frac{803972311}{1845632061}$, $\frac{2191643}{205070229}a^{14}-\frac{11225885}{2460842748}a^{13}-\frac{977605}{189295596}a^{12}-\frac{439884905}{2460842748}a^{11}+\frac{89844215}{205070229}a^{10}-\frac{957569537}{2460842748}a^{9}+\frac{197672455}{273426972}a^{8}-\frac{183962503}{189295596}a^{7}+\frac{795726457}{615210687}a^{6}+\frac{212929321}{189295596}a^{5}-\frac{6673891489}{2460842748}a^{4}+\frac{12880550701}{2460842748}a^{3}-\frac{3442246253}{615210687}a^{2}+\frac{2724360901}{615210687}a-\frac{847876532}{615210687}$, $\frac{5881345}{1845632061}a^{14}-\frac{878426}{1845632061}a^{13}-\frac{849617}{283943394}a^{12}-\frac{92207359}{1845632061}a^{11}+\frac{187507867}{1845632061}a^{10}-\frac{117970129}{1845632061}a^{9}+\frac{467014543}{3691264122}a^{8}+\frac{18133085}{141971697}a^{7}-\frac{69970627}{205070229}a^{6}+\frac{77863861}{141971697}a^{5}-\frac{714452759}{3691264122}a^{4}-\frac{113129615}{615210687}a^{3}+\frac{289828979}{615210687}a^{2}-\frac{306459256}{615210687}a+\frac{272231381}{1845632061}$, $\frac{55750519}{3691264122}a^{14}+\frac{83789396}{1845632061}a^{13}+\frac{7775311}{141971697}a^{12}-\frac{780378733}{3691264122}a^{11}-\frac{1042997537}{3691264122}a^{10}+\frac{61676869}{1845632061}a^{9}+\frac{1317639004}{1845632061}a^{8}+\frac{683040983}{283943394}a^{7}+\frac{2969650849}{1230421374}a^{6}+\frac{831907070}{141971697}a^{5}+\frac{6397648528}{1845632061}a^{4}+\frac{7324774727}{1230421374}a^{3}+\frac{591507037}{68356743}a^{2}+\frac{821281988}{205070229}a+\frac{10395023869}{1845632061}$, $\frac{40792931}{1230421374}a^{14}+\frac{38512711}{820280916}a^{13}+\frac{515931}{7010948}a^{12}-\frac{1311806999}{2460842748}a^{11}+\frac{160574332}{615210687}a^{10}-\frac{2077658909}{2460842748}a^{9}+\frac{7261057259}{2460842748}a^{8}+\frac{43913153}{63098532}a^{7}+\frac{7938843529}{1230421374}a^{6}+\frac{232118737}{63098532}a^{5}+\frac{3197488993}{820280916}a^{4}+\frac{25590076157}{2460842748}a^{3}+\frac{4302507127}{1230421374}a^{2}+\frac{5138844023}{615210687}a+\frac{1057745392}{615210687}$ Copy content Toggle raw display (assuming GRH)
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:  \( 2260889.53066 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 2260889.53066 \cdot 1}{2\cdot\sqrt{123256838893351061072896}}\cr\approx \mathstrut & 2.48961910823 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76)
 
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 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76, 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 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76);
 
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 + x^13 - 18*x^12 + 32*x^11 - 50*x^10 + 129*x^9 - 114*x^8 + 232*x^7 - 146*x^6 + 73*x^5 + 238*x^4 - 249*x^3 + 270*x^2 - 232*x + 76);
 
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

$D_{15}$ (as 15T2):

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 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.2956.1, 5.1.546121.1

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

Galois closure: deg 30
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 ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $15$ ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $15$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $15$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(739\) Copy content Toggle raw display $\Q_{739}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$