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Properties

Label 15.1.356...536.1
Degree $15$
Signature $[1, 7]$
Discriminant $-3.566\times 10^{22}$
Root discriminant \(31.88\)
Ramified primes $2,619$
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 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121)
 
gp: K = bnfinit(y^15 - 2*y^14 - 3*y^13 - 11*y^12 + 57*y^11 + 47*y^10 + 40*y^9 - 99*y^8 + 159*y^7 + 754*y^6 + 1337*y^5 + 1473*y^4 + 1003*y^3 + 669*y^2 + 286*y + 121, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121)
 

\( x^{15} - 2 x^{14} - 3 x^{13} - 11 x^{12} + 57 x^{11} + 47 x^{10} + 40 x^{9} - 99 x^{8} + 159 x^{7} + \cdots + 121 \) 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:   \(-35656154243640332225536\) \(\medspace = -\,2^{10}\cdot 619^{7}\) 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:  \(31.88\)
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:  $2^{2/3}619^{1/2}\approx 39.49407879378696$
Ramified primes:   \(2\), \(619\) 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:  \(\Q(\sqrt{-619}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a$, $\frac{1}{36}a^{11}+\frac{1}{18}a^{10}-\frac{1}{36}a^{9}+\frac{1}{36}a^{8}+\frac{1}{18}a^{7}-\frac{5}{36}a^{6}-\frac{11}{36}a^{5}+\frac{7}{18}a^{4}-\frac{17}{36}a^{3}+\frac{11}{36}a^{2}-\frac{5}{18}a+\frac{13}{36}$, $\frac{1}{1188}a^{12}+\frac{1}{1188}a^{11}-\frac{1}{396}a^{10}+\frac{8}{297}a^{9}-\frac{179}{1188}a^{8}+\frac{41}{1188}a^{7}+\frac{7}{99}a^{6}+\frac{481}{1188}a^{5}+\frac{437}{1188}a^{4}+\frac{47}{594}a^{3}+\frac{7}{36}a^{2}+\frac{179}{1188}a-\frac{29}{108}$, $\frac{1}{7128}a^{13}-\frac{1}{1782}a^{11}+\frac{35}{7128}a^{10}+\frac{185}{7128}a^{9}+\frac{5}{162}a^{8}-\frac{353}{7128}a^{7}-\frac{1187}{7128}a^{6}-\frac{23}{81}a^{5}-\frac{2323}{7128}a^{4}-\frac{1447}{7128}a^{3}+\frac{185}{1782}a^{2}-\frac{83}{1188}a-\frac{43}{648}$, $\frac{1}{722629512}a^{14}+\frac{35927}{722629512}a^{13}+\frac{7969}{90328689}a^{12}+\frac{1347877}{240876504}a^{11}+\frac{319637}{10948932}a^{10}+\frac{24133283}{722629512}a^{9}+\frac{8878129}{240876504}a^{8}+\frac{90541}{10948932}a^{7}+\frac{6002243}{80292168}a^{6}+\frac{336655141}{722629512}a^{5}+\frac{2085728}{30109563}a^{4}+\frac{22726139}{80292168}a^{3}-\frac{6070229}{32846796}a^{2}+\frac{336227317}{722629512}a-\frac{13052657}{65693592}$ 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:   $\frac{112505}{13382028}a^{14}-\frac{40255}{1824822}a^{13}-\frac{353407}{20073042}a^{12}-\frac{2447167}{40146084}a^{11}+\frac{20745361}{40146084}a^{10}+\frac{120718}{1115169}a^{9}-\frac{177201}{1486892}a^{8}-\frac{22659995}{40146084}a^{7}+\frac{17724422}{10036521}a^{6}+\frac{69637201}{13382028}a^{5}+\frac{89965015}{13382028}a^{4}+\frac{4604452}{912411}a^{3}+\frac{71773787}{20073042}a^{2}+\frac{56030609}{40146084}a+\frac{1454959}{1824822}$, $\frac{18940915}{722629512}a^{14}-\frac{22588655}{361314756}a^{13}-\frac{30109943}{361314756}a^{12}-\frac{39014225}{240876504}a^{11}+\frac{375386597}{240876504}a^{10}+\frac{13538899}{16423398}a^{9}-\frac{275644037}{240876504}a^{8}-\frac{327760655}{240876504}a^{7}+\frac{49054750}{10036521}a^{6}+\frac{13920776971}{722629512}a^{5}+\frac{5122289209}{240876504}a^{4}+\frac{151988746}{10036521}a^{3}+\frac{3364432535}{361314756}a^{2}+\frac{3073582123}{722629512}a+\frac{63920753}{32846796}$, $\frac{345799}{60219126}a^{14}-\frac{157163}{10948932}a^{13}-\frac{600871}{60219126}a^{12}-\frac{327577}{5474466}a^{11}+\frac{42723155}{120438252}a^{10}+\frac{4452925}{40146084}a^{9}+\frac{12551477}{60219126}a^{8}-\frac{86937017}{120438252}a^{7}+\frac{101144575}{120438252}a^{6}+\frac{76910233}{20073042}a^{5}+\frac{68293627}{10948932}a^{4}+\frac{650932283}{120438252}a^{3}+\frac{34714582}{30109563}a^{2}-\frac{7668472}{30109563}a-\frac{7538483}{10948932}$, $\frac{3287377}{240876504}a^{14}-\frac{6975913}{120438252}a^{13}+\frac{1381987}{30109563}a^{12}-\frac{28692263}{240876504}a^{11}+\frac{252776839}{240876504}a^{10}-\frac{13238729}{10036521}a^{9}+\frac{151962269}{240876504}a^{8}-\frac{449696605}{240876504}a^{7}+\frac{313269971}{60219126}a^{6}+\frac{280147825}{80292168}a^{5}-\frac{87722381}{240876504}a^{4}-\frac{92455018}{30109563}a^{3}-\frac{190142378}{30109563}a^{2}+\frac{592326727}{240876504}a-\frac{12921557}{2737233}$, $\frac{1942843}{722629512}a^{14}+\frac{603748}{90328689}a^{13}-\frac{16864829}{361314756}a^{12}-\frac{854429}{26764056}a^{11}+\frac{15101395}{240876504}a^{10}+\frac{337089133}{361314756}a^{9}-\frac{53020585}{240876504}a^{8}-\frac{55396993}{240876504}a^{7}-\frac{63104983}{120438252}a^{6}+\frac{3946503631}{722629512}a^{5}+\frac{754480897}{80292168}a^{4}+\frac{1065610237}{120438252}a^{3}+\frac{2670764069}{361314756}a^{2}+\frac{1722856807}{722629512}a+\frac{42333835}{16423398}$, $\frac{358900}{10036521}a^{14}-\frac{1504589}{20073042}a^{13}-\frac{104945}{912411}a^{12}-\frac{2044183}{6691014}a^{11}+\frac{13217533}{6691014}a^{10}+\frac{15881951}{10036521}a^{9}+\frac{1066655}{6691014}a^{8}-\frac{8733667}{6691014}a^{7}+\frac{5649277}{1115169}a^{6}+\frac{527240663}{20073042}a^{5}+\frac{274131707}{6691014}a^{4}+\frac{53021827}{1115169}a^{3}+\frac{777774161}{20073042}a^{2}+\frac{18114482}{912411}a+\frac{5365480}{912411}$, $\frac{1451665}{60219126}a^{14}-\frac{2881798}{30109563}a^{13}+\frac{3031067}{60219126}a^{12}-\frac{21916039}{120438252}a^{11}+\frac{54595553}{30109563}a^{10}-\frac{8195707}{4460676}a^{9}+\frac{40617913}{120438252}a^{8}-\frac{93843686}{30109563}a^{7}+\frac{1067884759}{120438252}a^{6}+\frac{108640117}{13382028}a^{5}+\frac{21357122}{30109563}a^{4}-\frac{790204225}{120438252}a^{3}-\frac{1685956139}{120438252}a^{2}+\frac{1132147}{2737233}a-\frac{74770415}{10948932}$ 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:  \( 1075964.47617 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 1075964.47617 \cdot 1}{2\cdot\sqrt{35656154243640332225536}}\cr\approx \mathstrut & 2.20287453016 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121)
 
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 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121, 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 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121);
 
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 - 3*x^13 - 11*x^12 + 57*x^11 + 47*x^10 + 40*x^9 - 99*x^8 + 159*x^7 + 754*x^6 + 1337*x^5 + 1473*x^4 + 1003*x^3 + 669*x^2 + 286*x + 121);
 
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

3.1.2476.1, 5.1.383161.1

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 ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ $15$ ${\href{/padicField/7.5.0.1}{5} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $15$ ${\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/padicField/43.2.0.1}{2} }^{7}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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}$
\(619\) Copy content Toggle raw display $\Q_{619}$$x$$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}$