# Properties

 Label 15.3.132...125.1 Degree $15$ Signature $[3, 6]$ Discriminant $1.325\times 10^{18}$ Root discriminant $$16.15$$ Ramified primes $5,7$ 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 - 14*x^10 + 35*x^5 + 7)

gp: K = bnfinit(y^15 - 14*y^10 + 35*y^5 + 7, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 14*x^10 + 35*x^5 + 7);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 14*x^10 + 35*x^5 + 7)

$$x^{15} - 14x^{10} + 35x^{5} + 7$$

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: $$1324654439158203125$$ 1324654439158203125 $$\medspace = 5^{9}\cdot 7^{14}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$16.15$$ 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: $5^{3/4}7^{14/15}\approx 20.558227794588174$ Ramified primes: $$5$$, $$7$$ 5, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{5})$$ $\card{ \Aut(K/\Q) }$: $3$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41}a^{10}+\frac{19}{41}a^{5}+\frac{6}{41}$, $\frac{1}{41}a^{11}+\frac{19}{41}a^{6}+\frac{6}{41}a$, $\frac{1}{205}a^{12}-\frac{1}{205}a^{11}+\frac{1}{205}a^{10}+\frac{2}{5}a^{9}+\frac{19}{205}a^{7}+\frac{22}{205}a^{6}-\frac{22}{205}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{47}{205}a^{2}+\frac{7}{41}a+\frac{6}{205}$, $\frac{1}{205}a^{13}-\frac{2}{205}a^{10}+\frac{2}{5}a^{9}+\frac{19}{205}a^{8}+\frac{1}{5}a^{7}-\frac{38}{205}a^{5}+\frac{1}{5}a^{4}-\frac{76}{205}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{94}{205}$, $\frac{1}{205}a^{14}-\frac{2}{205}a^{11}+\frac{2}{205}a^{10}+\frac{19}{205}a^{9}+\frac{1}{5}a^{8}-\frac{38}{205}a^{6}-\frac{44}{205}a^{5}-\frac{76}{205}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{94}{205}a-\frac{14}{41}$

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: $\frac{3}{41}a^{10}-\frac{25}{41}a^{5}-\frac{23}{41}$, $\frac{3}{41}a^{10}-\frac{25}{41}a^{5}+\frac{18}{41}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}-\frac{1}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}+\frac{22}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a+\frac{76}{205}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}-\frac{1}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}+\frac{22}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a-\frac{129}{205}$, $\frac{2}{41}a^{14}-\frac{13}{205}a^{13}+\frac{4}{205}a^{12}+\frac{1}{205}a^{11}+\frac{1}{41}a^{10}-\frac{138}{205}a^{9}+\frac{163}{205}a^{8}-\frac{47}{205}a^{7}-\frac{22}{205}a^{6}-\frac{69}{205}a^{5}+\frac{388}{205}a^{4}-\frac{324}{205}a^{3}+\frac{147}{205}a^{2}+\frac{47}{205}a+\frac{71}{205}$, $\frac{2}{205}a^{14}-\frac{8}{205}a^{13}-\frac{3}{205}a^{12}+\frac{4}{205}a^{11}+\frac{2}{205}a^{10}-\frac{44}{205}a^{9}+\frac{27}{41}a^{8}+\frac{5}{41}a^{7}-\frac{47}{205}a^{6}-\frac{3}{205}a^{5}+\frac{258}{205}a^{4}-\frac{499}{205}a^{3}+\frac{21}{41}a^{2}+\frac{24}{205}a-\frac{111}{205}$, $\frac{4}{205}a^{14}-\frac{1}{41}a^{13}+\frac{6}{205}a^{12}+\frac{6}{205}a^{11}+\frac{4}{205}a^{10}-\frac{47}{205}a^{9}+\frac{69}{205}a^{8}-\frac{91}{205}a^{7}-\frac{10}{41}a^{6}-\frac{88}{205}a^{5}+\frac{13}{41}a^{4}-\frac{47}{41}a^{3}+\frac{241}{205}a^{2}-\frac{46}{205}a+\frac{106}{205}$, $\frac{1}{41}a^{14}-\frac{12}{205}a^{13}+\frac{3}{41}a^{12}-\frac{1}{205}a^{10}-\frac{69}{205}a^{9}+\frac{182}{205}a^{8}-\frac{207}{205}a^{7}-\frac{19}{205}a^{5}+\frac{153}{205}a^{4}-\frac{523}{205}a^{3}+\frac{541}{205}a^{2}-\frac{2}{5}a-\frac{47}{205}$ 3/41*a^10 - 25/41*a^5 - 23/41, 3/41*a^10 - 25/41*a^5 + 18/41, 4/205*a^14 - 1/41*a^13 + 6/205*a^12 + 6/205*a^11 - 1/205*a^10 - 47/205*a^9 + 69/205*a^8 - 91/205*a^7 - 10/41*a^6 + 22/205*a^5 + 13/41*a^4 - 47/41*a^3 + 241/205*a^2 - 46/205*a + 76/205, 4/205*a^14 - 1/41*a^13 + 6/205*a^12 + 6/205*a^11 - 1/205*a^10 - 47/205*a^9 + 69/205*a^8 - 91/205*a^7 - 10/41*a^6 + 22/205*a^5 + 13/41*a^4 - 47/41*a^3 + 241/205*a^2 - 46/205*a - 129/205, 2/41*a^14 - 13/205*a^13 + 4/205*a^12 + 1/205*a^11 + 1/41*a^10 - 138/205*a^9 + 163/205*a^8 - 47/205*a^7 - 22/205*a^6 - 69/205*a^5 + 388/205*a^4 - 324/205*a^3 + 147/205*a^2 + 47/205*a + 71/205, 2/205*a^14 - 8/205*a^13 - 3/205*a^12 + 4/205*a^11 + 2/205*a^10 - 44/205*a^9 + 27/41*a^8 + 5/41*a^7 - 47/205*a^6 - 3/205*a^5 + 258/205*a^4 - 499/205*a^3 + 21/41*a^2 + 24/205*a - 111/205, 4/205*a^14 - 1/41*a^13 + 6/205*a^12 + 6/205*a^11 + 4/205*a^10 - 47/205*a^9 + 69/205*a^8 - 91/205*a^7 - 10/41*a^6 - 88/205*a^5 + 13/41*a^4 - 47/41*a^3 + 241/205*a^2 - 46/205*a + 106/205, 1/41*a^14 - 12/205*a^13 + 3/41*a^12 - 1/205*a^10 - 69/205*a^9 + 182/205*a^8 - 207/205*a^7 - 19/205*a^5 + 153/205*a^4 - 523/205*a^3 + 541/205*a^2 - 2/5*a - 47/205 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: $$1272.85837366$$ 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 1272.85837366 \cdot 1}{2\cdot\sqrt{1324654439158203125}}\cr\approx \mathstrut & 0.272187380385 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^15 - 14*x^10 + 35*x^5 + 7)

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 - 14*x^10 + 35*x^5 + 7, 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 - 14*x^10 + 35*x^5 + 7);

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 - 14*x^10 + 35*x^5 + 7);

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 ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ R R $15$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ $15$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.5.0.1}{5} }^{3}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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]])