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Normalized defining polynomial

\( x^{15} - 3x - 4 \) x^15 - 3*x - 4

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:		\(-117386800991476702805311488\) -117386800991476702805311488 \(\medspace = -\,2^{14}\cdot 3^{15}\cdot 71\cdot 859\cdot 881\cdot 9292939\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(54.70\)	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:		not computed
Ramified primes:		\(2\), \(3\), \(71\), \(859\), \(881\), \(9292939\) 2, 3, 71, 859, 881, 9292939	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		$\Q(\sqrt{-14979\!\cdots\!81453}$)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

Unit group

Rank:		$7$	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:		$a^{14}-a^{12}+a^{10}+2a^{7}+a^{6}-2a^{5}-a^{4}+2a^{3}+a^{2}+1$, $a^{11}+a^{9}+a^{8}-2a^{4}-a^{3}-a^{2}-3a-1$, $a^{14}+a^{13}-2a^{11}+2a^{10}-2a^{8}-a^{7}+4a^{6}-3a^{4}+a^{3}+3a^{2}-4a-7$, $a^{13}-a^{12}-a^{11}+2a^{10}-2a^{8}+a^{7}+a^{6}-a^{5}-1$, $a^{11}-2a^{10}+2a^{9}-2a^{8}-a^{5}+a^{2}-a+3$, $2a^{14}+8a^{13}-6a^{12}-8a^{11}+10a^{10}+5a^{9}-14a^{8}-2a^{7}+16a^{6}-3a^{5}-20a^{4}+8a^{3}+22a^{2}-20a-29$, $9a^{14}-3a^{13}-10a^{12}+16a^{11}-16a^{10}+13a^{9}-6a^{8}+2a^{7}+17a^{6}-24a^{5}+26a^{4}-11a^{3}-3a^{2}+4a-51$ a^14 - a^12 + a^10 + 2*a^7 + a^6 - 2*a^5 - a^4 + 2*a^3 + a^2 + 1, a^11 + a^9 + a^8 - 2*a^4 - a^3 - a^2 - 3*a - 1, a^14 + a^13 - 2*a^11 + 2*a^10 - 2*a^8 - a^7 + 4*a^6 - 3*a^4 + a^3 + 3*a^2 - 4*a - 7, a^13 - a^12 - a^11 + 2*a^10 - 2*a^8 + a^7 + a^6 - a^5 - 1, a^11 - 2*a^10 + 2*a^9 - 2*a^8 - a^5 + a^2 - a + 3, 2*a^14 + 8*a^13 - 6*a^12 - 8*a^11 + 10*a^10 + 5*a^9 - 14*a^8 - 2*a^7 + 16*a^6 - 3*a^5 - 20*a^4 + 8*a^3 + 22*a^2 - 20*a - 29, 9*a^14 - 3*a^13 - 10*a^12 + 16*a^11 - 16*a^10 + 13*a^9 - 6*a^8 + 2*a^7 + 17*a^6 - 24*a^5 + 26*a^4 - 11*a^3 - 3*a^2 + 4*a - 51 (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:		\( 24861699.7289 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 24861699.7289 \cdot 2}{2\cdot\sqrt{117386800991476702805311488}}\cr\approx \mathstrut & 1.77423138488 \end{aligned}\] (assuming GRH)

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

 

x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x - 4)

 

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 - 3*x - 4, 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 - 3*x - 4);

 

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 - 3*x - 4);

 

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

$S_{15}$ (as 15T104):

A non-solvable group of order 1307674368000

The 176 conjugacy class representatives for $S_{15}$

Character table for $S_{15}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.4.0.1}{4} }$	${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	$15$	${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.6.0.1}{6} }$	${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.7.0.1}{7} }$	${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$	$15$	${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.2.2.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.6.6.5	$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
	2.6.6.3	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
\(3\) 3	3.3.3.1	$x^{3} + 6 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
	3.12.12.11	$x^{12} - 48 x^{11} + 846 x^{10} - 5820 x^{9} + 38376 x^{8} + 37098 x^{7} - 36666 x^{6} - 17928 x^{5} + 11340 x^{4} - 1188 x^{3} + 486 x + 81$	$3$	$4$	$12$	12T170	$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
\(71\) 71	71.2.1.1	$x^{2} + 497$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.2.0.1	$x^{2} + 69 x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.11.0.1	$x^{11} + 48 x + 64$	$1$	$11$	$0$	$C_{11}$	$[\ ]^{11}$
\(859\) 859		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $9$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
\(881\) 881	$\Q_{881}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
\(9292939\) 9292939		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $9$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$

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