LMFDB - Number field 21.1.34154868120917133312000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16)

gp: K = bnfinit(y^21 - 10*y^18 + 36*y^15 - 76*y^12 - 224*y^9 - 144*y^6 - 32*y^3 - 16, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16)

$ x^{21} - 10x^{18} + 36x^{15} - 76x^{12} - 224x^{9} - 144x^{6} - 32x^{3} - 16 $ x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$21$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$34154868120917133312000000000$ 34154868120917133312000000000 $\medspace = 2^{20}\cdot 3^{34}\cdot 5^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$22.84$	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:	$2^{20/21}3^{31/18}5^{1/2}\approx 28.700473627493697$
Ramified primes:	$2$, $3$, $5$ 2, 3, 5	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}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{1}{18}a^{7}+\frac{2}{9}a^{6}+\frac{1}{9}a^{4}+\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{36}a^{11}+\frac{1}{18}a^{9}+\frac{1}{9}a^{8}-\frac{1}{6}a^{7}+\frac{1}{18}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{3}-\frac{1}{9}a^{2}-\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{36}a^{12}+\frac{1}{18}a^{9}-\frac{1}{6}a^{6}-\frac{2}{9}a^{3}+\frac{2}{9}$, $\frac{1}{36}a^{13}-\frac{1}{18}a^{9}+\frac{1}{6}a^{8}-\frac{2}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{3}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a^{2}-\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{36}a^{14}+\frac{1}{18}a^{9}+\frac{1}{9}a^{8}-\frac{1}{6}a^{7}+\frac{1}{18}a^{6}-\frac{1}{3}a^{5}+\frac{1}{9}a^{3}+\frac{4}{9}a^{2}-\frac{1}{3}a+\frac{4}{9}$, $\frac{1}{108}a^{15}-\frac{1}{27}a^{9}-\frac{2}{27}a^{6}-\frac{1}{3}a^{3}+\frac{8}{27}$, $\frac{1}{648}a^{16}-\frac{1}{324}a^{15}-\frac{1}{108}a^{14}+\frac{1}{108}a^{12}+\frac{1}{108}a^{11}-\frac{1}{162}a^{10}-\frac{13}{162}a^{9}+\frac{1}{6}a^{8}-\frac{29}{162}a^{7}+\frac{2}{81}a^{6}-\frac{10}{27}a^{5}+\frac{4}{9}a^{4}-\frac{5}{27}a^{3}+\frac{13}{27}a^{2}-\frac{23}{81}a-\frac{20}{81}$, $\frac{1}{648}a^{17}+\frac{1}{324}a^{15}+\frac{1}{108}a^{14}+\frac{1}{108}a^{13}+\frac{1}{81}a^{11}+\frac{1}{54}a^{10}+\frac{7}{162}a^{9}-\frac{11}{162}a^{8}+\frac{1}{9}a^{7}+\frac{16}{81}a^{6}+\frac{10}{27}a^{5}-\frac{2}{27}a^{4}+\frac{37}{81}a^{2}-\frac{7}{27}a-\frac{10}{81}$, $\frac{1}{648}a^{18}-\frac{1}{324}a^{15}-\frac{1}{162}a^{12}+\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{31}{162}a^{6}+\frac{22}{81}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{26}{81}$, $\frac{1}{648}a^{19}+\frac{1}{324}a^{15}+\frac{1}{108}a^{14}-\frac{1}{162}a^{13}-\frac{1}{108}a^{12}-\frac{1}{108}a^{11}-\frac{1}{81}a^{10}+\frac{2}{81}a^{9}+\frac{1}{6}a^{8}-\frac{2}{9}a^{7}-\frac{13}{162}a^{6}+\frac{10}{27}a^{5}+\frac{4}{81}a^{4}+\frac{2}{27}a^{3}+\frac{5}{27}a^{2}-\frac{1}{3}a-\frac{16}{81}$, $\frac{1}{648}a^{20}-\frac{1}{324}a^{15}+\frac{1}{81}a^{14}-\frac{1}{108}a^{13}-\frac{1}{324}a^{11}-\frac{1}{54}a^{10}-\frac{7}{162}a^{9}+\frac{2}{9}a^{8}+\frac{1}{18}a^{7}+\frac{11}{81}a^{6}+\frac{28}{81}a^{5}+\frac{2}{27}a^{4}-\frac{2}{27}a^{2}-\frac{11}{27}a-\frac{17}{81}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6, 1/2*a^7, 1/2*a^8, 1/6*a^9 + 1/6*a^6 + 1/3*a^3 + 1/3, 1/18*a^10 + 1/18*a^9 - 1/6*a^8 + 1/18*a^7 + 2/9*a^6 + 1/9*a^4 + 1/9*a^3 - 1/3*a^2 + 4/9*a - 2/9, 1/36*a^11 + 1/18*a^9 + 1/9*a^8 - 1/6*a^7 + 1/18*a^6 - 4/9*a^5 + 1/9*a^3 - 1/9*a^2 - 1/3*a + 4/9, 1/36*a^12 + 1/18*a^9 - 1/6*a^6 - 2/9*a^3 + 2/9, 1/36*a^13 - 1/18*a^9 + 1/6*a^8 - 2/9*a^7 - 2/9*a^6 - 1/3*a^4 - 1/9*a^3 + 1/3*a^2 - 2/9*a + 2/9, 1/36*a^14 + 1/18*a^9 + 1/9*a^8 - 1/6*a^7 + 1/18*a^6 - 1/3*a^5 + 1/9*a^3 + 4/9*a^2 - 1/3*a + 4/9, 1/108*a^15 - 1/27*a^9 - 2/27*a^6 - 1/3*a^3 + 8/27, 1/648*a^16 - 1/324*a^15 - 1/108*a^14 + 1/108*a^12 + 1/108*a^11 - 1/162*a^10 - 13/162*a^9 + 1/6*a^8 - 29/162*a^7 + 2/81*a^6 - 10/27*a^5 + 4/9*a^4 - 5/27*a^3 + 13/27*a^2 - 23/81*a - 20/81, 1/648*a^17 + 1/324*a^15 + 1/108*a^14 + 1/108*a^13 + 1/81*a^11 + 1/54*a^10 + 7/162*a^9 - 11/162*a^8 + 1/9*a^7 + 16/81*a^6 + 10/27*a^5 - 2/27*a^4 + 37/81*a^2 - 7/27*a - 10/81, 1/648*a^18 - 1/324*a^15 - 1/162*a^12 + 1/18*a^9 - 1/6*a^8 + 1/6*a^7 + 31/162*a^6 + 22/81*a^3 - 1/3*a^2 + 1/3*a - 26/81, 1/648*a^19 + 1/324*a^15 + 1/108*a^14 - 1/162*a^13 - 1/108*a^12 - 1/108*a^11 - 1/81*a^10 + 2/81*a^9 + 1/6*a^8 - 2/9*a^7 - 13/162*a^6 + 10/27*a^5 + 4/81*a^4 + 2/27*a^3 + 5/27*a^2 - 1/3*a - 16/81, 1/648*a^20 - 1/324*a^15 + 1/81*a^14 - 1/108*a^13 - 1/324*a^11 - 1/54*a^10 - 7/162*a^9 + 2/9*a^8 + 1/18*a^7 + 11/81*a^6 + 28/81*a^5 + 2/27*a^4 - 2/27*a^2 - 11/27*a - 17/81

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 $1$ (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:	$10$	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:	$\frac{19}{648}a^{20}-\frac{1}{324}a^{19}-\frac{1}{216}a^{18}-\frac{49}{162}a^{17}+\frac{5}{216}a^{16}+\frac{7}{162}a^{15}+\frac{373}{324}a^{14}-\frac{11}{324}a^{13}-\frac{7}{54}a^{12}-\frac{421}{162}a^{11}-\frac{4}{81}a^{10}+\frac{29}{162}a^{9}-\frac{461}{81}a^{8}+\frac{71}{54}a^{7}+\frac{241}{162}a^{6}-\frac{206}{81}a^{5}+\frac{175}{81}a^{4}+\frac{19}{27}a^{3}-\frac{34}{81}a^{2}+\frac{43}{27}a+\frac{7}{81}$, $\frac{41}{648}a^{20}+\frac{5}{648}a^{19}-\frac{1}{81}a^{18}-\frac{101}{162}a^{17}-\frac{29}{324}a^{16}+\frac{43}{324}a^{15}+\frac{707}{324}a^{14}+\frac{131}{324}a^{13}-\frac{173}{324}a^{12}-\frac{1441}{324}a^{11}-\frac{19}{18}a^{10}+\frac{100}{81}a^{9}-\frac{2423}{162}a^{8}-\frac{59}{81}a^{7}+\frac{359}{162}a^{6}-\frac{901}{81}a^{5}+\frac{125}{81}a^{4}-\frac{68}{81}a^{3}-\frac{182}{81}a^{2}+\frac{68}{81}a-\frac{61}{81}$, $\frac{2}{81}a^{20}+\frac{1}{324}a^{19}-\frac{11}{648}a^{18}-\frac{41}{162}a^{17}-\frac{1}{36}a^{16}+\frac{55}{324}a^{15}+\frac{307}{324}a^{14}+\frac{23}{324}a^{13}-\frac{50}{81}a^{12}-\frac{223}{108}a^{11}-\frac{2}{81}a^{10}+\frac{221}{162}a^{9}-\frac{421}{81}a^{8}-\frac{4}{3}a^{7}+\frac{563}{162}a^{6}-\frac{155}{81}a^{5}-\frac{10}{81}a^{4}+\frac{274}{81}a^{3}+\frac{40}{81}a^{2}+\frac{2}{9}a+\frac{95}{81}$, $\frac{4}{81}a^{20}-\frac{23}{324}a^{19}-\frac{23}{648}a^{18}-\frac{155}{324}a^{17}+\frac{461}{648}a^{16}+\frac{113}{324}a^{15}+\frac{265}{162}a^{14}-\frac{413}{162}a^{13}-\frac{395}{324}a^{12}-\frac{361}{108}a^{11}+\frac{47}{9}a^{10}+\frac{5}{2}a^{9}-\frac{1873}{162}a^{8}+\frac{2705}{162}a^{7}+\frac{674}{81}a^{6}-\frac{1006}{81}a^{5}+\frac{617}{81}a^{4}+\frac{556}{81}a^{3}-\frac{367}{81}a^{2}-\frac{100}{81}a+\frac{211}{81}$, $\frac{17}{324}a^{20}-\frac{1}{324}a^{19}+\frac{7}{216}a^{18}-\frac{335}{648}a^{17}+\frac{1}{36}a^{16}-\frac{107}{324}a^{15}+\frac{583}{324}a^{14}-\frac{23}{324}a^{13}+\frac{11}{9}a^{12}-\frac{43}{12}a^{11}+\frac{2}{81}a^{10}-\frac{425}{162}a^{9}-\frac{2075}{162}a^{8}+\frac{4}{3}a^{7}-\frac{569}{81}a^{6}-\frac{671}{81}a^{5}+\frac{10}{81}a^{4}-\frac{8}{3}a^{3}-\frac{11}{81}a^{2}-\frac{11}{9}a+\frac{65}{81}$, $\frac{11}{648}a^{20}+\frac{7}{162}a^{19}-\frac{53}{324}a^{17}-\frac{287}{648}a^{16}+\frac{44}{81}a^{14}+\frac{269}{162}a^{13}-\frac{109}{108}a^{11}-\frac{98}{27}a^{10}-\frac{362}{81}a^{8}-\frac{1469}{162}a^{7}-\frac{280}{81}a^{5}-\frac{266}{81}a^{4}+\frac{53}{81}a^{2}+\frac{4}{81}a+1$, $\frac{5}{162}a^{20}+\frac{1}{72}a^{19}-\frac{1}{54}a^{18}-\frac{221}{648}a^{17}-\frac{4}{27}a^{16}+\frac{31}{162}a^{15}+\frac{235}{162}a^{14}+\frac{65}{108}a^{13}-\frac{79}{108}a^{12}-\frac{298}{81}a^{11}-\frac{40}{27}a^{10}+\frac{269}{162}a^{9}-\frac{611}{162}a^{8}-\frac{55}{27}a^{7}+\frac{571}{162}a^{6}+\frac{104}{81}a^{5}-\frac{28}{27}a^{4}+\frac{47}{27}a^{3}+\frac{64}{81}a^{2}-\frac{1}{27}a+\frac{91}{81}$, $\frac{19}{648}a^{20}+\frac{7}{648}a^{19}-\frac{1}{216}a^{18}-\frac{49}{162}a^{17}-\frac{23}{216}a^{16}+\frac{7}{162}a^{15}+\frac{373}{324}a^{14}+\frac{31}{81}a^{13}-\frac{7}{54}a^{12}-\frac{421}{162}a^{11}-\frac{70}{81}a^{10}+\frac{29}{162}a^{9}-\frac{461}{81}a^{8}-\frac{58}{27}a^{7}+\frac{241}{162}a^{6}-\frac{206}{81}a^{5}-\frac{230}{81}a^{4}+\frac{19}{27}a^{3}-\frac{34}{81}a^{2}-\frac{10}{9}a+\frac{7}{81}$, $\frac{1}{81}a^{20}+\frac{5}{324}a^{19}-\frac{1}{81}a^{18}-\frac{95}{648}a^{17}-\frac{19}{108}a^{16}+\frac{47}{324}a^{15}+\frac{227}{324}a^{14}+\frac{64}{81}a^{13}-\frac{221}{324}a^{12}-\frac{661}{324}a^{11}-\frac{175}{81}a^{10}+\frac{35}{18}a^{9}+\frac{19}{162}a^{8}-\frac{47}{54}a^{7}+\frac{37}{162}a^{6}+\frac{50}{81}a^{5}+\frac{10}{81}a^{4}-\frac{80}{81}a^{3}-\frac{47}{81}a^{2}+\frac{26}{27}a+\frac{73}{81}$, $\frac{7}{216}a^{20}+\frac{1}{27}a^{19}-\frac{5}{162}a^{18}-\frac{215}{648}a^{17}-\frac{127}{324}a^{16}+\frac{107}{324}a^{15}+\frac{34}{27}a^{14}+\frac{14}{9}a^{13}-\frac{217}{162}a^{12}-\frac{941}{324}a^{11}-\frac{296}{81}a^{10}+\frac{59}{18}a^{9}-\frac{487}{81}a^{8}-\frac{520}{81}a^{7}+\frac{733}{162}a^{6}-\frac{125}{27}a^{5}-a^{4}+\frac{112}{81}a^{3}-\frac{149}{81}a^{2}+\frac{34}{81}a-\frac{47}{81}$ 19/648a^20 - 1/324a^19 - 1/216a^18 - 49/162a^17 + 5/216a^16 + 7/162a^15 + 373/324a^14 - 11/324a^13 - 7/54a^12 - 421/162a^11 - 4/81a^10 + 29/162a^9 - 461/81a^8 + 71/54a^7 + 241/162a^6 - 206/81a^5 + 175/81a^4 + 19/27a^3 - 34/81a^2 + 43/27a + 7/81, 41/648a^20 + 5/648a^19 - 1/81a^18 - 101/162a^17 - 29/324a^16 + 43/324a^15 + 707/324a^14 + 131/324a^13 - 173/324a^12 - 1441/324a^11 - 19/18a^10 + 100/81a^9 - 2423/162a^8 - 59/81a^7 + 359/162a^6 - 901/81a^5 + 125/81a^4 - 68/81a^3 - 182/81a^2 + 68/81a - 61/81, 2/81a^20 + 1/324a^19 - 11/648a^18 - 41/162a^17 - 1/36a^16 + 55/324a^15 + 307/324a^14 + 23/324a^13 - 50/81a^12 - 223/108a^11 - 2/81a^10 + 221/162a^9 - 421/81a^8 - 4/3a^7 + 563/162a^6 - 155/81a^5 - 10/81a^4 + 274/81a^3 + 40/81a^2 + 2/9a + 95/81, 4/81a^20 - 23/324a^19 - 23/648a^18 - 155/324a^17 + 461/648a^16 + 113/324a^15 + 265/162a^14 - 413/162a^13 - 395/324a^12 - 361/108a^11 + 47/9a^10 + 5/2a^9 - 1873/162a^8 + 2705/162a^7 + 674/81a^6 - 1006/81a^5 + 617/81a^4 + 556/81a^3 - 367/81a^2 - 100/81a + 211/81, 17/324a^20 - 1/324a^19 + 7/216a^18 - 335/648a^17 + 1/36a^16 - 107/324a^15 + 583/324a^14 - 23/324a^13 + 11/9a^12 - 43/12a^11 + 2/81a^10 - 425/162a^9 - 2075/162a^8 + 4/3a^7 - 569/81a^6 - 671/81a^5 + 10/81a^4 - 8/3a^3 - 11/81a^2 - 11/9a + 65/81, 11/648a^20 + 7/162a^19 - 53/324a^17 - 287/648a^16 + 44/81a^14 + 269/162a^13 - 109/108a^11 - 98/27a^10 - 362/81a^8 - 1469/162a^7 - 280/81a^5 - 266/81a^4 + 53/81a^2 + 4/81a + 1, 5/162a^20 + 1/72a^19 - 1/54a^18 - 221/648a^17 - 4/27a^16 + 31/162a^15 + 235/162a^14 + 65/108a^13 - 79/108a^12 - 298/81a^11 - 40/27a^10 + 269/162a^9 - 611/162a^8 - 55/27a^7 + 571/162a^6 + 104/81a^5 - 28/27a^4 + 47/27a^3 + 64/81a^2 - 1/27a + 91/81, 19/648a^20 + 7/648a^19 - 1/216a^18 - 49/162a^17 - 23/216a^16 + 7/162a^15 + 373/324a^14 + 31/81a^13 - 7/54a^12 - 421/162a^11 - 70/81a^10 + 29/162a^9 - 461/81a^8 - 58/27a^7 + 241/162a^6 - 206/81a^5 - 230/81a^4 + 19/27a^3 - 34/81a^2 - 10/9a + 7/81, 1/81a^20 + 5/324a^19 - 1/81a^18 - 95/648a^17 - 19/108a^16 + 47/324a^15 + 227/324a^14 + 64/81a^13 - 221/324a^12 - 661/324a^11 - 175/81a^10 + 35/18a^9 + 19/162a^8 - 47/54a^7 + 37/162a^6 + 50/81a^5 + 10/81a^4 - 80/81a^3 - 47/81a^2 + 26/27a + 73/81, 7/216a^20 + 1/27a^19 - 5/162a^18 - 215/648a^17 - 127/324a^16 + 107/324a^15 + 34/27a^14 + 14/9a^13 - 217/162a^12 - 941/324a^11 - 296/81a^10 + 59/18a^9 - 487/81a^8 - 520/81a^7 + 733/162a^6 - 125/27a^5 - a^4 + 112/81a^3 - 149/81a^2 + 34/81*a - 47/81 (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:	$ 2085380.41806 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 2085380.41806 \cdot 1}{2\cdot\sqrt{34154868120917133312000000000}}\cr\approx \mathstrut & 1.08207576000 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16)

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^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16, 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^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16);

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^21 - 10*x^18 + 36*x^15 - 76*x^12 - 224*x^9 - 144*x^6 - 32*x^3 - 16);

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

$S_3\times F_7$ (as 21T15):

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 252

The 21 conjugacy class representatives for $S_3\times F_7$

Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.108.1, 7.1.157464000.1

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

Degree 42 siblings:	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	R	R	${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$	${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$	$21$	${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$	${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$	${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }$	${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.21.20.1	$x^{21} + 2$	$21$	$1$	$20$	21T11	$[\ ]_{21}^{6}$
$3$ 3	3.3.3.2	$x^{3} + 3 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
$3$ 3		Deg $18$	$18$	$1$	$31$
$5$ 5	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.6.3.2	$x^{6} + 75 x^{2} - 375$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.12.6.1	$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$

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