LMFDB - Number field 20.0.34088221436915805032899514033281.2

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625)

gp: K = bnfinit(y^20 - y^19 - 4*y^18 + 9*y^17 + 11*y^16 - 56*y^15 + y^14 + 279*y^13 - 284*y^12 - 1111*y^11 + 2531*y^10 - 5555*y^9 - 7100*y^8 + 34875*y^7 + 625*y^6 - 175000*y^5 + 171875*y^4 + 703125*y^3 - 1562500*y^2 - 1953125*y + 9765625, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625)

$ x^{20} - x^{19} - 4 x^{18} + 9 x^{17} + 11 x^{16} - 56 x^{15} + x^{14} + 279 x^{13} - 284 x^{12} + \cdots + 9765625 $ x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$20$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$34088221436915805032899514033281$ 34088221436915805032899514033281 $\medspace = 11^{18}\cdot 19^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$37.73$	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:	$11^{9/10}19^{1/2}\approx 37.72508414373865$
Ramified primes:	$11$, $19$ 11, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$20$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$209=11\cdot 19$
Dirichlet character group:	$\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(134,·)$, $\chi_{209}(75,·)$, $\chi_{209}(208,·)$, $\chi_{209}(18,·)$, $\chi_{209}(20,·)$, $\chi_{209}(151,·)$, $\chi_{209}(153,·)$, $\chi_{209}(94,·)$, $\chi_{209}(96,·)$, $\chi_{209}(37,·)$, $\chi_{209}(39,·)$, $\chi_{209}(170,·)$, $\chi_{209}(172,·)$, $\chi_{209}(113,·)$, $\chi_{209}(115,·)$, $\chi_{209}(56,·)$, $\chi_{209}(58,·)$, $\chi_{209}(189,·)$, $\chi_{209}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{512}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{12655}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1111}{2531}$, $\frac{1}{63275}a^{12}-\frac{1}{63275}a^{11}-\frac{9}{25}a^{10}-\frac{11}{25}a^{9}+\frac{6}{25}a^{8}-\frac{1}{25}a^{7}-\frac{4}{25}a^{6}+\frac{9}{25}a^{5}+\frac{11}{25}a^{4}-\frac{6}{25}a^{3}+\frac{1}{25}a^{2}-\frac{1111}{12655}a-\frac{284}{2531}$, $\frac{1}{316375}a^{13}-\frac{1}{316375}a^{12}-\frac{4}{316375}a^{11}-\frac{36}{125}a^{10}-\frac{44}{125}a^{9}-\frac{26}{125}a^{8}-\frac{4}{125}a^{7}+\frac{9}{125}a^{6}+\frac{11}{125}a^{5}-\frac{56}{125}a^{4}+\frac{1}{125}a^{3}-\frac{1111}{63275}a^{2}-\frac{284}{12655}a+\frac{279}{2531}$, $\frac{1}{1581875}a^{14}-\frac{1}{1581875}a^{13}-\frac{4}{1581875}a^{12}+\frac{9}{1581875}a^{11}+\frac{81}{625}a^{10}+\frac{99}{625}a^{9}+\frac{121}{625}a^{8}+\frac{9}{625}a^{7}+\frac{11}{625}a^{6}-\frac{56}{625}a^{5}+\frac{1}{625}a^{4}-\frac{1111}{316375}a^{3}-\frac{284}{63275}a^{2}+\frac{279}{12655}a+\frac{1}{2531}$, $\frac{1}{7909375}a^{15}-\frac{1}{7909375}a^{14}-\frac{4}{7909375}a^{13}+\frac{9}{7909375}a^{12}+\frac{11}{7909375}a^{11}-\frac{526}{3125}a^{10}+\frac{121}{3125}a^{9}-\frac{616}{3125}a^{8}+\frac{11}{3125}a^{7}-\frac{56}{3125}a^{6}+\frac{1}{3125}a^{5}-\frac{1111}{1581875}a^{4}-\frac{284}{316375}a^{3}+\frac{279}{63275}a^{2}+\frac{1}{12655}a-\frac{56}{2531}$, $\frac{1}{39546875}a^{16}-\frac{1}{39546875}a^{15}-\frac{4}{39546875}a^{14}+\frac{9}{39546875}a^{13}+\frac{11}{39546875}a^{12}-\frac{56}{39546875}a^{11}+\frac{6371}{15625}a^{10}-\frac{3741}{15625}a^{9}+\frac{3136}{15625}a^{8}-\frac{56}{15625}a^{7}+\frac{1}{15625}a^{6}-\frac{1111}{7909375}a^{5}-\frac{284}{1581875}a^{4}+\frac{279}{316375}a^{3}+\frac{1}{63275}a^{2}-\frac{56}{12655}a+\frac{11}{2531}$, $\frac{1}{197734375}a^{17}-\frac{1}{197734375}a^{16}-\frac{4}{197734375}a^{15}+\frac{9}{197734375}a^{14}+\frac{11}{197734375}a^{13}-\frac{56}{197734375}a^{12}+\frac{1}{197734375}a^{11}+\frac{11884}{78125}a^{10}+\frac{34386}{78125}a^{9}-\frac{15681}{78125}a^{8}+\frac{1}{78125}a^{7}-\frac{1111}{39546875}a^{6}-\frac{284}{7909375}a^{5}+\frac{279}{1581875}a^{4}+\frac{1}{316375}a^{3}-\frac{56}{63275}a^{2}+\frac{11}{12655}a+\frac{9}{2531}$, $\frac{1}{988671875}a^{18}-\frac{1}{988671875}a^{17}-\frac{4}{988671875}a^{16}+\frac{9}{988671875}a^{15}+\frac{11}{988671875}a^{14}-\frac{56}{988671875}a^{13}+\frac{1}{988671875}a^{12}+\frac{279}{988671875}a^{11}+\frac{112511}{390625}a^{10}-\frac{171931}{390625}a^{9}+\frac{1}{390625}a^{8}-\frac{1111}{197734375}a^{7}-\frac{284}{39546875}a^{6}+\frac{279}{7909375}a^{5}+\frac{1}{1581875}a^{4}-\frac{56}{316375}a^{3}+\frac{11}{63275}a^{2}+\frac{9}{12655}a-\frac{4}{2531}$, $\frac{1}{4943359375}a^{19}-\frac{1}{4943359375}a^{18}-\frac{4}{4943359375}a^{17}+\frac{9}{4943359375}a^{16}+\frac{11}{4943359375}a^{15}-\frac{56}{4943359375}a^{14}+\frac{1}{4943359375}a^{13}+\frac{279}{4943359375}a^{12}-\frac{284}{4943359375}a^{11}-\frac{562556}{1953125}a^{10}+\frac{1}{1953125}a^{9}-\frac{1111}{988671875}a^{8}-\frac{284}{197734375}a^{7}+\frac{279}{39546875}a^{6}+\frac{1}{7909375}a^{5}-\frac{56}{1581875}a^{4}+\frac{11}{316375}a^{3}+\frac{9}{63275}a^{2}-\frac{4}{12655}a-\frac{1}{2531}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, 1/12655*a^11 - 1/5*a^10 + 1/5*a^9 - 1/5*a^8 + 1/5*a^7 - 1/5*a^6 + 1/5*a^5 - 1/5*a^4 + 1/5*a^3 - 1/5*a^2 + 1/5*a - 1111/2531, 1/63275*a^12 - 1/63275*a^11 - 9/25*a^10 - 11/25*a^9 + 6/25*a^8 - 1/25*a^7 - 4/25*a^6 + 9/25*a^5 + 11/25*a^4 - 6/25*a^3 + 1/25*a^2 - 1111/12655*a - 284/2531, 1/316375*a^13 - 1/316375*a^12 - 4/316375*a^11 - 36/125*a^10 - 44/125*a^9 - 26/125*a^8 - 4/125*a^7 + 9/125*a^6 + 11/125*a^5 - 56/125*a^4 + 1/125*a^3 - 1111/63275*a^2 - 284/12655*a + 279/2531, 1/1581875*a^14 - 1/1581875*a^13 - 4/1581875*a^12 + 9/1581875*a^11 + 81/625*a^10 + 99/625*a^9 + 121/625*a^8 + 9/625*a^7 + 11/625*a^6 - 56/625*a^5 + 1/625*a^4 - 1111/316375*a^3 - 284/63275*a^2 + 279/12655*a + 1/2531, 1/7909375*a^15 - 1/7909375*a^14 - 4/7909375*a^13 + 9/7909375*a^12 + 11/7909375*a^11 - 526/3125*a^10 + 121/3125*a^9 - 616/3125*a^8 + 11/3125*a^7 - 56/3125*a^6 + 1/3125*a^5 - 1111/1581875*a^4 - 284/316375*a^3 + 279/63275*a^2 + 1/12655*a - 56/2531, 1/39546875*a^16 - 1/39546875*a^15 - 4/39546875*a^14 + 9/39546875*a^13 + 11/39546875*a^12 - 56/39546875*a^11 + 6371/15625*a^10 - 3741/15625*a^9 + 3136/15625*a^8 - 56/15625*a^7 + 1/15625*a^6 - 1111/7909375*a^5 - 284/1581875*a^4 + 279/316375*a^3 + 1/63275*a^2 - 56/12655*a + 11/2531, 1/197734375*a^17 - 1/197734375*a^16 - 4/197734375*a^15 + 9/197734375*a^14 + 11/197734375*a^13 - 56/197734375*a^12 + 1/197734375*a^11 + 11884/78125*a^10 + 34386/78125*a^9 - 15681/78125*a^8 + 1/78125*a^7 - 1111/39546875*a^6 - 284/7909375*a^5 + 279/1581875*a^4 + 1/316375*a^3 - 56/63275*a^2 + 11/12655*a + 9/2531, 1/988671875*a^18 - 1/988671875*a^17 - 4/988671875*a^16 + 9/988671875*a^15 + 11/988671875*a^14 - 56/988671875*a^13 + 1/988671875*a^12 + 279/988671875*a^11 + 112511/390625*a^10 - 171931/390625*a^9 + 1/390625*a^8 - 1111/197734375*a^7 - 284/39546875*a^6 + 279/7909375*a^5 + 1/1581875*a^4 - 56/316375*a^3 + 11/63275*a^2 + 9/12655*a - 4/2531, 1/4943359375*a^19 - 1/4943359375*a^18 - 4/4943359375*a^17 + 9/4943359375*a^16 + 11/4943359375*a^15 - 56/4943359375*a^14 + 1/4943359375*a^13 + 279/4943359375*a^12 - 284/4943359375*a^11 - 562556/1953125*a^10 + 1/1953125*a^9 - 1111/988671875*a^8 - 284/197734375*a^7 + 279/39546875*a^6 + 1/7909375*a^5 - 56/1581875*a^4 + 11/316375*a^3 + 9/63275*a^2 - 4/12655*a - 1/2531

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

$C_{55}$, which has order $55$ (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:	$9$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{56}{39546875} a^{17} + \frac{308549}{39546875} a^{6} $ -(56)/(39546875)a^(17) + (308549)/(39546875)a^(6) (order $22$)	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:	$\frac{1111}{4943359375}a^{19}-\frac{1136}{4943359375}a^{18}+\frac{2556}{4943359375}a^{17}+\frac{9999}{4943359375}a^{16}+\frac{12221}{4943359375}a^{15}+\frac{284}{4943359375}a^{14}+\frac{79236}{4943359375}a^{13}+\frac{309969}{4943359375}a^{12}-\frac{315524}{4943359375}a^{11}+\frac{284}{1953125}a^{10}+\frac{1111}{1953125}a^{9}-\frac{1234321}{988671875}a^{8}+\frac{79236}{39546875}a^{7}+\frac{284}{7909375}a^{6}+\frac{1111}{7909375}a^{5}-\frac{62216}{1581875}a^{4}+\frac{2556}{63275}a^{3}-\frac{1136}{12655}a^{2}-\frac{4444}{12655}a-\frac{1111}{2531}$, $\frac{284}{4943359375}a^{19}+\frac{1111}{4943359375}a^{18}-\frac{2556}{4943359375}a^{17}-\frac{9999}{4943359375}a^{16}+\frac{15904}{4943359375}a^{15}+\frac{62216}{4943359375}a^{14}-\frac{79236}{4943359375}a^{13}-\frac{309969}{4943359375}a^{12}+\frac{315524}{4943359375}a^{11}-\frac{284}{1953125}a^{10}-\frac{1111}{1953125}a^{9}+\frac{80656}{197734375}a^{8}+\frac{315524}{197734375}a^{7}-\frac{284}{7909375}a^{6}-\frac{1111}{7909375}a^{5}-\frac{3124}{316375}a^{4}-\frac{12221}{316375}a^{3}+\frac{1136}{12655}a^{2}+\frac{4444}{12655}a-\frac{1420}{2531}$, $\frac{1}{197734375}a^{18}-\frac{56}{39546875}a^{17}-\frac{1}{63275}a^{13}+\frac{1}{12655}a^{12}-\frac{711704}{197734375}a^{7}+\frac{308549}{39546875}a^{6}+\frac{15679}{63275}a^{2}-\frac{3024}{12655}a$, $\frac{284}{4943359375}a^{19}+\frac{1136}{4943359375}a^{18}+\frac{4444}{4943359375}a^{17}-\frac{3124}{4943359375}a^{16}-\frac{12221}{4943359375}a^{15}+\frac{62216}{4943359375}a^{14}-\frac{79236}{4943359375}a^{13}-\frac{309969}{4943359375}a^{12}+\frac{315524}{4943359375}a^{11}-\frac{284}{1953125}a^{10}-\frac{1111}{1953125}a^{9}+\frac{80656}{197734375}a^{8}-\frac{79236}{39546875}a^{7}-\frac{309969}{39546875}a^{6}+\frac{15904}{1581875}a^{5}+\frac{62216}{1581875}a^{4}-\frac{12221}{316375}a^{3}+\frac{1136}{12655}a^{2}+\frac{4444}{12655}a-\frac{1420}{2531}$, $\frac{2556}{4943359375}a^{19}+\frac{11444}{4943359375}a^{18}+\frac{10526}{4943359375}a^{17}-\frac{47496}{4943359375}a^{16}-\frac{111384}{4943359375}a^{15}-\frac{1136}{4943359375}a^{14}-\frac{4444}{4943359375}a^{13}-\frac{68001}{4943359375}a^{12}+\frac{1262096}{4943359375}a^{11}-\frac{1136}{1953125}a^{10}-\frac{4444}{1953125}a^{9}-\frac{2776441}{988671875}a^{8}-\frac{280649}{197734375}a^{7}+\frac{492081}{39546875}a^{6}+\frac{626629}{7909375}a^{5}+\frac{246069}{1581875}a^{4}-\frac{10224}{63275}a^{3}-\frac{39996}{63275}a^{2}+\frac{8704}{12655}a+\frac{1913}{2531}$, $\frac{6173}{4943359375}a^{19}-\frac{13248}{4943359375}a^{18}-\frac{3692}{4943359375}a^{17}-\frac{6943}{4943359375}a^{16}+\frac{130403}{4943359375}a^{15}-\frac{283188}{4943359375}a^{14}+\frac{396798}{4943359375}a^{13}+\frac{159767}{4943359375}a^{12}-\frac{1753132}{4943359375}a^{11}-\frac{1938}{1953125}a^{10}+\frac{6173}{1953125}a^{9}-\frac{6858203}{988671875}a^{8}+\frac{76989}{7909375}a^{7}+\frac{159324}{7909375}a^{6}-\frac{7802}{7909375}a^{5}-\frac{342893}{1581875}a^{4}+\frac{68462}{316375}a^{3}+\frac{40437}{63275}a^{2}-\frac{12596}{12655}a+\frac{3951}{2531}$, $\frac{1704}{4943359375}a^{19}-\frac{4494}{4943359375}a^{18}+\frac{23999}{4943359375}a^{17}-\frac{9029}{4943359375}a^{16}-\frac{124716}{4943359375}a^{15}+\frac{1111}{4943359375}a^{14}+\frac{466219}{4943359375}a^{13}+\frac{75101}{4943359375}a^{12}-\frac{1234321}{4943359375}a^{11}+\frac{1111}{1953125}a^{10}-\frac{2531}{1953125}a^{9}-\frac{6798222}{988671875}a^{8}+\frac{2973253}{197734375}a^{7}-\frac{611543}{39546875}a^{6}+\frac{158731}{7909375}a^{5}+\frac{11662}{316375}a^{4}+\frac{9999}{63275}a^{3}-\frac{53578}{63275}a^{2}-\frac{8579}{12655}a+\frac{10617}{2531}$, $\frac{334}{4943359375}a^{19}+\frac{6666}{4943359375}a^{18}-\frac{1236}{4943359375}a^{17}+\frac{44656}{4943359375}a^{16}-\frac{52851}{4943359375}a^{15}+\frac{107696}{4943359375}a^{14}-\frac{77816}{4943359375}a^{13}-\frac{695039}{4943359375}a^{12}-\frac{87756}{4943359375}a^{11}-\frac{1679}{1953125}a^{10}+\frac{309}{1953125}a^{9}-\frac{3901819}{988671875}a^{8}-\frac{918797}{197734375}a^{7}+\frac{86211}{39546875}a^{6}-\frac{227609}{7909375}a^{5}+\frac{138368}{1581875}a^{4}+\frac{4517}{316375}a^{3}+\frac{3692}{12655}a^{2}+\frac{4812}{12655}a+\frac{4753}{2531}$, $\frac{5012}{4943359375}a^{19}-\frac{3592}{4943359375}a^{18}-\frac{21568}{4943359375}a^{17}+\frac{81528}{4943359375}a^{16}-\frac{22438}{4943359375}a^{15}-\frac{344577}{4943359375}a^{14}+\frac{550517}{4943359375}a^{13}+\frac{625493}{4943359375}a^{12}-\frac{1034328}{4943359375}a^{11}-\frac{827}{1953125}a^{10}+\frac{3642}{1953125}a^{9}-\frac{1318783}{988671875}a^{8}-\frac{2457736}{197734375}a^{7}+\frac{1324667}{39546875}a^{6}-\frac{28729}{1581875}a^{5}-\frac{206747}{1581875}a^{4}+\frac{117898}{316375}a^{3}-\frac{548}{2531}a^{2}-\frac{11544}{12655}a+\frac{1420}{2531}$ 1111/4943359375a^19 - 1136/4943359375a^18 + 2556/4943359375a^17 + 9999/4943359375a^16 + 12221/4943359375a^15 + 284/4943359375a^14 + 79236/4943359375a^13 + 309969/4943359375a^12 - 315524/4943359375a^11 + 284/1953125a^10 + 1111/1953125a^9 - 1234321/988671875a^8 + 79236/39546875a^7 + 284/7909375a^6 + 1111/7909375a^5 - 62216/1581875a^4 + 2556/63275a^3 - 1136/12655a^2 - 4444/12655a - 1111/2531, 284/4943359375a^19 + 1111/4943359375a^18 - 2556/4943359375a^17 - 9999/4943359375a^16 + 15904/4943359375a^15 + 62216/4943359375a^14 - 79236/4943359375a^13 - 309969/4943359375a^12 + 315524/4943359375a^11 - 284/1953125a^10 - 1111/1953125a^9 + 80656/197734375a^8 + 315524/197734375a^7 - 284/7909375a^6 - 1111/7909375a^5 - 3124/316375a^4 - 12221/316375a^3 + 1136/12655a^2 + 4444/12655a - 1420/2531, 1/197734375a^18 - 56/39546875a^17 - 1/63275a^13 + 1/12655a^12 - 711704/197734375a^7 + 308549/39546875a^6 + 15679/63275a^2 - 3024/12655a, 284/4943359375a^19 + 1136/4943359375a^18 + 4444/4943359375a^17 - 3124/4943359375a^16 - 12221/4943359375a^15 + 62216/4943359375a^14 - 79236/4943359375a^13 - 309969/4943359375a^12 + 315524/4943359375a^11 - 284/1953125a^10 - 1111/1953125a^9 + 80656/197734375a^8 - 79236/39546875a^7 - 309969/39546875a^6 + 15904/1581875a^5 + 62216/1581875a^4 - 12221/316375a^3 + 1136/12655a^2 + 4444/12655a - 1420/2531, 2556/4943359375a^19 + 11444/4943359375a^18 + 10526/4943359375a^17 - 47496/4943359375a^16 - 111384/4943359375a^15 - 1136/4943359375a^14 - 4444/4943359375a^13 - 68001/4943359375a^12 + 1262096/4943359375a^11 - 1136/1953125a^10 - 4444/1953125a^9 - 2776441/988671875a^8 - 280649/197734375a^7 + 492081/39546875a^6 + 626629/7909375a^5 + 246069/1581875a^4 - 10224/63275a^3 - 39996/63275a^2 + 8704/12655a + 1913/2531, 6173/4943359375a^19 - 13248/4943359375a^18 - 3692/4943359375a^17 - 6943/4943359375a^16 + 130403/4943359375a^15 - 283188/4943359375a^14 + 396798/4943359375a^13 + 159767/4943359375a^12 - 1753132/4943359375a^11 - 1938/1953125a^10 + 6173/1953125a^9 - 6858203/988671875a^8 + 76989/7909375a^7 + 159324/7909375a^6 - 7802/7909375a^5 - 342893/1581875a^4 + 68462/316375a^3 + 40437/63275a^2 - 12596/12655a + 3951/2531, 1704/4943359375a^19 - 4494/4943359375a^18 + 23999/4943359375a^17 - 9029/4943359375a^16 - 124716/4943359375a^15 + 1111/4943359375a^14 + 466219/4943359375a^13 + 75101/4943359375a^12 - 1234321/4943359375a^11 + 1111/1953125a^10 - 2531/1953125a^9 - 6798222/988671875a^8 + 2973253/197734375a^7 - 611543/39546875a^6 + 158731/7909375a^5 + 11662/316375a^4 + 9999/63275a^3 - 53578/63275a^2 - 8579/12655a + 10617/2531, 334/4943359375a^19 + 6666/4943359375a^18 - 1236/4943359375a^17 + 44656/4943359375a^16 - 52851/4943359375a^15 + 107696/4943359375a^14 - 77816/4943359375a^13 - 695039/4943359375a^12 - 87756/4943359375a^11 - 1679/1953125a^10 + 309/1953125a^9 - 3901819/988671875a^8 - 918797/197734375a^7 + 86211/39546875a^6 - 227609/7909375a^5 + 138368/1581875a^4 + 4517/316375a^3 + 3692/12655a^2 + 4812/12655a + 4753/2531, 5012/4943359375a^19 - 3592/4943359375a^18 - 21568/4943359375a^17 + 81528/4943359375a^16 - 22438/4943359375a^15 - 344577/4943359375a^14 + 550517/4943359375a^13 + 625493/4943359375a^12 - 1034328/4943359375a^11 - 827/1953125a^10 + 3642/1953125a^9 - 1318783/988671875a^8 - 2457736/197734375a^7 + 1324667/39546875a^6 - 28729/1581875a^5 - 206747/1581875a^4 + 117898/316375a^3 - 548/2531a^2 - 11544/12655a + 1420/2531 (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:	$ 8845130.03478 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 8845130.03478 \cdot 55}{22\cdot\sqrt{34088221436915805032899514033281}}\cr\approx \mathstrut & 0.363195737573 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625)

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^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625, 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^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625);

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^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625);

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

$C_2\times C_{10}$ (as 20T3):

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)

An abelian group of order 20

The 20 conjugacy class representatives for $C_2\times C_{10}$

Character table for $C_2\times C_{10}$

Intermediate fields

$\Q(\sqrt{209}) $, $\Q(\sqrt{-11}) $, $\Q(\sqrt{-19}) $, $\Q(\sqrt{-11}, \sqrt{-19})$, $\Q(\zeta_{11})^+$, 10.10.5838511919737409.1, $\Q(\zeta_{11})$, 10.0.530773810885219.1

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.10.0.1}{10} }^{2}$	${\href{/padicField/3.10.0.1}{10} }^{2}$	${\href{/padicField/5.5.0.1}{5} }^{4}$	${\href{/padicField/7.10.0.1}{10} }^{2}$	R	${\href{/padicField/13.10.0.1}{10} }^{2}$	${\href{/padicField/17.10.0.1}{10} }^{2}$	R	${\href{/padicField/23.1.0.1}{1} }^{20}$	${\href{/padicField/29.10.0.1}{10} }^{2}$	${\href{/padicField/31.10.0.1}{10} }^{2}$	${\href{/padicField/37.10.0.1}{10} }^{2}$	${\href{/padicField/41.10.0.1}{10} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{10}$	${\href{/padicField/47.5.0.1}{5} }^{4}$	${\href{/padicField/53.10.0.1}{10} }^{2}$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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
$11$ 11	11.10.9.7	$x^{10} + 11$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
$11$ 11	11.10.9.7	$x^{10} + 11$	$10$	$1$	$9$	$C_{10}$	$[\ ]_{10}$
$19$ 19	19.20.10.1	$x^{20} + 190 x^{18} + 16245 x^{16} + 36 x^{15} + 823106 x^{14} - 3386 x^{13} + 27361982 x^{12} - 466480 x^{11} + 623608258 x^{10} - 16361762 x^{9} + 9874880623 x^{8} - 202428566 x^{7} + 107361060449 x^{6} + 371628878 x^{5} + 766799662247 x^{4} + 26518316394 x^{3} + 3240176666731 x^{2} + 149940122806 x + 6108844895300$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$

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