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Properties

Label 15.1.202...384.1
Degree $15$
Signature $[1, 7]$
Discriminant $-2.027\times 10^{22}$
Root discriminant \(30.70\)
Ramified primes $2,571$
Class number $2$
Class group [2]
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 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243)
 
gp: K = bnfinit(y^15 - 7*y^14 + 20*y^13 - 32*y^12 + 27*y^11 + 5*y^10 + 21*y^9 - 267*y^8 + 615*y^7 - 795*y^6 + 1025*y^5 - 1443*y^4 + 1530*y^3 - 1188*y^2 + 729*y - 243, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243)
 

\( x^{15} - 7 x^{14} + 20 x^{13} - 32 x^{12} + 27 x^{11} + 5 x^{10} + 21 x^{9} - 267 x^{8} + 615 x^{7} + \cdots - 243 \) 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:   \(-20265284932691591056384\) \(\medspace = -\,2^{10}\cdot 571^{7}\) 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:  \(30.70\)
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:  $2^{2/3}571^{1/2}\approx 37.93191056327041$
Ramified primes:   \(2\), \(571\) 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:  \(\Q(\sqrt{-571}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}$, $\frac{1}{18}a^{10}-\frac{1}{9}a^{8}-\frac{1}{18}a^{7}-\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{1}{6}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{6}a$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{9}a^{7}+\frac{1}{18}a^{6}-\frac{1}{6}a^{5}+\frac{4}{9}a^{4}-\frac{1}{18}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{54}a^{12}+\frac{1}{54}a^{11}+\frac{1}{54}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{8}-\frac{1}{54}a^{7}-\frac{5}{54}a^{6}+\frac{17}{54}a^{5}-\frac{23}{54}a^{4}-\frac{25}{54}a^{3}-\frac{7}{18}a^{2}+\frac{1}{6}a$, $\frac{1}{324}a^{13}-\frac{1}{162}a^{12}+\frac{1}{324}a^{11}-\frac{1}{36}a^{10}-\frac{1}{12}a^{9}-\frac{49}{324}a^{8}-\frac{35}{324}a^{7}-\frac{1}{324}a^{6}-\frac{47}{324}a^{5}+\frac{5}{324}a^{4}-\frac{13}{108}a^{3}+\frac{17}{36}a^{2}+\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{242676}a^{14}+\frac{97}{242676}a^{13}-\frac{161}{34668}a^{12}-\frac{815}{40446}a^{11}+\frac{128}{20223}a^{10}+\frac{53}{60669}a^{9}-\frac{12559}{121338}a^{8}-\frac{9931}{60669}a^{7}+\frac{7129}{60669}a^{6}+\frac{4763}{60669}a^{5}+\frac{7961}{40446}a^{4}-\frac{2563}{40446}a^{3}-\frac{113}{2996}a^{2}-\frac{1031}{2996}a+\frac{641}{2996}$ 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

$C_{2}$, which has order $2$

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:   $\frac{13129}{242676}a^{14}-\frac{18859}{60669}a^{13}+\frac{24305}{34668}a^{12}-\frac{215987}{242676}a^{11}+\frac{32729}{80892}a^{10}+\frac{174581}{242676}a^{9}+\frac{163529}{80892}a^{8}-\frac{957041}{80892}a^{7}+\frac{1529845}{80892}a^{6}-\frac{1672795}{80892}a^{5}+\frac{7469867}{242676}a^{4}-\frac{1108805}{26964}a^{3}+\frac{464495}{13482}a^{2}-\frac{214427}{8988}a+\frac{8782}{749}$, $\frac{11909}{242676}a^{14}-\frac{58207}{242676}a^{13}+\frac{16355}{34668}a^{12}-\frac{11756}{20223}a^{11}+\frac{3133}{20223}a^{10}+\frac{31228}{60669}a^{9}+\frac{126797}{60669}a^{8}-\frac{509750}{60669}a^{7}+\frac{724394}{60669}a^{6}-\frac{1778779}{121338}a^{5}+\frac{891937}{40446}a^{4}-\frac{992693}{40446}a^{3}+\frac{179719}{8988}a^{2}-\frac{42515}{2996}a+\frac{14841}{2996}$, $\frac{2356}{60669}a^{14}-\frac{49835}{242676}a^{13}+\frac{6953}{17334}a^{12}-\frac{34333}{80892}a^{11}+\frac{5273}{80892}a^{10}+\frac{130631}{242676}a^{9}+\frac{429607}{242676}a^{8}-\frac{1871539}{242676}a^{7}+\frac{2283277}{242676}a^{6}-\frac{2238499}{242676}a^{5}+\frac{1434595}{80892}a^{4}-\frac{1751831}{80892}a^{3}+\frac{351277}{26964}a^{2}-\frac{21059}{2247}a+\frac{16577}{2996}$, $\frac{661}{20223}a^{14}-\frac{15419}{80892}a^{13}+\frac{407}{963}a^{12}-\frac{44113}{80892}a^{11}+\frac{26321}{80892}a^{10}+\frac{32483}{80892}a^{9}+\frac{105949}{80892}a^{8}-\frac{193819}{26964}a^{7}+\frac{97505}{8988}a^{6}-\frac{373171}{26964}a^{5}+\frac{1697903}{80892}a^{4}-\frac{1948865}{80892}a^{3}+\frac{619309}{26964}a^{2}-\frac{28617}{1498}a+\frac{24917}{2996}$, $\frac{3344}{60669}a^{14}-\frac{41093}{121338}a^{13}+\frac{6995}{8667}a^{12}-\frac{7073}{6741}a^{11}+\frac{11149}{20223}a^{10}+\frac{48310}{60669}a^{9}+\frac{110618}{60669}a^{8}-\frac{798842}{60669}a^{7}+\frac{1352012}{60669}a^{6}-\frac{1459853}{60669}a^{5}+\frac{238100}{6741}a^{4}-\frac{965191}{20223}a^{3}+\frac{89501}{2247}a^{2}-\frac{41229}{1498}a+\frac{10352}{749}$, $\frac{5875}{121338}a^{14}-\frac{35317}{121338}a^{13}+\frac{11561}{17334}a^{12}-\frac{50236}{60669}a^{11}+\frac{2749}{6741}a^{10}+\frac{43024}{60669}a^{9}+\frac{3740}{2247}a^{8}-\frac{228293}{20223}a^{7}+\frac{39580}{2247}a^{6}-\frac{43027}{2247}a^{5}+\frac{1871959}{60669}a^{4}-\frac{763709}{20223}a^{3}+\frac{396119}{13482}a^{2}-\frac{95009}{4494}a+\frac{11887}{1498}$, $\frac{1303}{34668}a^{14}-\frac{3947}{17334}a^{13}+\frac{18605}{34668}a^{12}-\frac{23681}{34668}a^{11}+\frac{4057}{11556}a^{10}+\frac{17189}{34668}a^{9}+\frac{4765}{3852}a^{8}-\frac{102677}{11556}a^{7}+\frac{167797}{11556}a^{6}-\frac{178117}{11556}a^{5}+\frac{867215}{34668}a^{4}-\frac{372491}{11556}a^{3}+\frac{48065}{1926}a^{2}-\frac{7391}{428}a+\frac{1649}{214}$ 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:  \( 420279.707763 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 420279.707763 \cdot 2}{2\cdot\sqrt{20265284932691591056384}}\cr\approx \mathstrut & 2.28271208790 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243)
 
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 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243, 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 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243);
 
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 - 7*x^14 + 20*x^13 - 32*x^12 + 27*x^11 + 5*x^10 + 21*x^9 - 267*x^8 + 615*x^7 - 795*x^6 + 1025*x^5 - 1443*x^4 + 1530*x^3 - 1188*x^2 + 729*x - 243);
 
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

3.1.2284.1, 5.1.326041.1

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 ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{7}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $15$ $15$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $15$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $15$

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}$
\(571\) Copy content Toggle raw display $\Q_{571}$$x$$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}$