# 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)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining 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)

$$x^{15} - 5 x^{14} + 16 x^{13} - 39 x^{12} + 65 x^{11} - 91 x^{10} + 78 x^{9} - 39 x^{8} - 26 x^{7} + 78 x^{6} - 78 x^{5} + 52 x^{4} - 13 x^{3} - 9 x^{2} + 6 x - 1$$

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: $$1240576436601868288$$ 1240576436601868288 $$\medspace = 2^{12}\cdot 13^{13}$$ 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: $$2$$, $$13$$ 2, 13 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}-\frac{4}{13}a^{11}+\frac{3}{13}a^{10}+\frac{1}{13}a^{9}-\frac{4}{13}a^{8}+\frac{3}{13}a^{7}+\frac{1}{13}a^{6}-\frac{4}{13}a^{5}+\frac{3}{13}a^{4}+\frac{1}{13}a^{3}-\frac{4}{13}a^{2}+\frac{3}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{4}{13}$, $\frac{1}{403}a^{14}+\frac{6}{403}a^{13}-\frac{11}{403}a^{12}+\frac{57}{403}a^{11}-\frac{176}{403}a^{10}-\frac{167}{403}a^{9}+\frac{70}{403}a^{8}-\frac{137}{403}a^{7}-\frac{76}{403}a^{6}-\frac{138}{403}a^{5}-\frac{46}{403}a^{4}+\frac{197}{403}a^{3}-\frac{47}{403}a^{2}-\frac{185}{403}a-\frac{13}{31}$

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

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]])