LMFDB - Number field 21.3.7146646609494406531041460224.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1)

gp: K = bnfinit(y^21 - 3*y^20 - 4*y^19 + 16*y^18 - 9*y^17 - 33*y^16 + 52*y^15 + 12*y^14 - 136*y^13 + 96*y^12 + 180*y^11 - 260*y^10 - 144*y^9 + 312*y^8 + 44*y^7 - 220*y^6 + 15*y^5 + 59*y^4 - 16*y^3 + 36*y^2 - 31*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1)

$ x^{21} - 3 x^{20} - 4 x^{19} + 16 x^{18} - 9 x^{17} - 33 x^{16} + 52 x^{15} + 12 x^{14} - 136 x^{13} + \cdots + 1 $ x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*x + 1

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:	$[3, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-7146646609494406531041460224$ -7146646609494406531041460224 $\medspace = -\,2^{64}\cdot 3^{18}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$21.20$	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:	$2^{29/8}3^{6/7}\approx 31.63696468726565$
Ramified primes:	$2$, $3$ 2, 3	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-1}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{8}a$, $\frac{1}{8}a^{16}-\frac{1}{8}$, $\frac{1}{64}a^{17}-\frac{3}{64}a^{16}+\frac{1}{32}a^{15}-\frac{1}{32}a^{14}+\frac{1}{16}a^{13}+\frac{7}{32}a^{11}+\frac{5}{32}a^{10}-\frac{1}{4}a^{9}-\frac{1}{16}a^{8}-\frac{15}{32}a^{7}-\frac{9}{32}a^{6}+\frac{3}{16}a^{5}-\frac{3}{8}a^{4}+\frac{11}{32}a^{3}+\frac{1}{32}a^{2}-\frac{1}{64}a+\frac{15}{64}$, $\frac{1}{64}a^{18}+\frac{1}{64}a^{16}-\frac{1}{16}a^{15}-\frac{1}{32}a^{14}+\frac{1}{16}a^{13}-\frac{1}{32}a^{12}+\frac{3}{16}a^{11}+\frac{7}{32}a^{10}+\frac{1}{16}a^{9}+\frac{3}{32}a^{8}+\frac{3}{16}a^{7}+\frac{11}{32}a^{6}+\frac{1}{16}a^{5}-\frac{1}{32}a^{4}+\frac{7}{16}a^{3}+\frac{5}{64}a^{2}+\frac{1}{16}a+\frac{21}{64}$, $\frac{1}{128}a^{19}-\frac{1}{128}a^{18}-\frac{1}{128}a^{17}-\frac{7}{128}a^{16}+\frac{3}{64}a^{15}+\frac{1}{64}a^{14}+\frac{5}{64}a^{13}-\frac{5}{64}a^{12}+\frac{7}{64}a^{11}-\frac{3}{64}a^{10}-\frac{3}{64}a^{9}-\frac{5}{64}a^{8}-\frac{25}{64}a^{7}-\frac{27}{64}a^{6}+\frac{29}{64}a^{5}+\frac{27}{64}a^{4}+\frac{37}{128}a^{3}+\frac{19}{128}a^{2}-\frac{21}{128}a-\frac{3}{128}$, $\frac{1}{1157298870656}a^{20}-\frac{1048673419}{1157298870656}a^{19}-\frac{779216257}{1157298870656}a^{18}+\frac{2580505063}{1157298870656}a^{17}-\frac{27905486493}{578649435328}a^{16}-\frac{9131520381}{578649435328}a^{15}+\frac{8596105857}{578649435328}a^{14}+\frac{11920676885}{578649435328}a^{13}+\frac{3984972003}{578649435328}a^{12}-\frac{50090446817}{578649435328}a^{11}+\frac{128354582921}{578649435328}a^{10}+\frac{66215511389}{578649435328}a^{9}+\frac{2597510291}{578649435328}a^{8}-\frac{59552884241}{578649435328}a^{7}+\frac{83339962201}{578649435328}a^{6}-\frac{243029498667}{578649435328}a^{5}-\frac{330362936963}{1157298870656}a^{4}-\frac{191820525839}{1157298870656}a^{3}-\frac{513957271741}{1157298870656}a^{2}-\frac{401219358285}{1157298870656}a-\frac{53116372489}{144662358832}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2, 1/2*a^9 - 1/2*a, 1/2*a^10 - 1/2*a^2, 1/2*a^11 - 1/2*a^3, 1/4*a^12 - 1/4*a^8 + 1/4*a^4 - 1/4, 1/4*a^13 - 1/4*a^9 + 1/4*a^5 - 1/4*a, 1/8*a^14 - 1/8*a^12 + 1/8*a^10 - 1/8*a^8 + 1/8*a^6 - 1/8*a^4 + 1/8*a^2 - 1/8, 1/8*a^15 - 1/8*a^13 + 1/8*a^11 - 1/8*a^9 + 1/8*a^7 - 1/8*a^5 + 1/8*a^3 - 1/8*a, 1/8*a^16 - 1/8, 1/64*a^17 - 3/64*a^16 + 1/32*a^15 - 1/32*a^14 + 1/16*a^13 + 7/32*a^11 + 5/32*a^10 - 1/4*a^9 - 1/16*a^8 - 15/32*a^7 - 9/32*a^6 + 3/16*a^5 - 3/8*a^4 + 11/32*a^3 + 1/32*a^2 - 1/64*a + 15/64, 1/64*a^18 + 1/64*a^16 - 1/16*a^15 - 1/32*a^14 + 1/16*a^13 - 1/32*a^12 + 3/16*a^11 + 7/32*a^10 + 1/16*a^9 + 3/32*a^8 + 3/16*a^7 + 11/32*a^6 + 1/16*a^5 - 1/32*a^4 + 7/16*a^3 + 5/64*a^2 + 1/16*a + 21/64, 1/128*a^19 - 1/128*a^18 - 1/128*a^17 - 7/128*a^16 + 3/64*a^15 + 1/64*a^14 + 5/64*a^13 - 5/64*a^12 + 7/64*a^11 - 3/64*a^10 - 3/64*a^9 - 5/64*a^8 - 25/64*a^7 - 27/64*a^6 + 29/64*a^5 + 27/64*a^4 + 37/128*a^3 + 19/128*a^2 - 21/128*a - 3/128, 1/1157298870656*a^20 - 1048673419/1157298870656*a^19 - 779216257/1157298870656*a^18 + 2580505063/1157298870656*a^17 - 27905486493/578649435328*a^16 - 9131520381/578649435328*a^15 + 8596105857/578649435328*a^14 + 11920676885/578649435328*a^13 + 3984972003/578649435328*a^12 - 50090446817/578649435328*a^11 + 128354582921/578649435328*a^10 + 66215511389/578649435328*a^9 + 2597510291/578649435328*a^8 - 59552884241/578649435328*a^7 + 83339962201/578649435328*a^6 - 243029498667/578649435328*a^5 - 330362936963/1157298870656*a^4 - 191820525839/1157298870656*a^3 - 513957271741/1157298870656*a^2 - 401219358285/1157298870656*a - 53116372489/144662358832

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

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:	$11$	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:	$\frac{129427364769}{1157298870656}a^{20}-\frac{158973451327}{578649435328}a^{19}-\frac{351863426395}{578649435328}a^{18}+\frac{213741308075}{144662358832}a^{17}-\frac{147170050183}{1157298870656}a^{16}-\frac{139141792923}{36165589708}a^{15}+\frac{133303594185}{36165589708}a^{14}+\frac{1065014444933}{289324717664}a^{13}-\frac{3862091276047}{289324717664}a^{12}+\frac{448956981469}{144662358832}a^{11}+\frac{1651205609303}{72331179416}a^{10}-\frac{4704181620609}{289324717664}a^{9}-\frac{7740895406949}{289324717664}a^{8}+\frac{750934815771}{36165589708}a^{7}+\frac{2688136179979}{144662358832}a^{6}-\frac{4202981530531}{289324717664}a^{5}-\frac{8711131285301}{1157298870656}a^{4}+\frac{2051237676699}{578649435328}a^{3}+\frac{375446281071}{578649435328}a^{2}+\frac{1194438179365}{289324717664}a-\frac{748174762069}{1157298870656}$, $\frac{83285702587}{578649435328}a^{20}-\frac{471299973371}{1157298870656}a^{19}-\frac{731066570167}{1157298870656}a^{18}+\frac{2508421152449}{1157298870656}a^{17}-\frac{1162863783073}{1157298870656}a^{16}-\frac{2769265833775}{578649435328}a^{15}+\frac{3859091009665}{578649435328}a^{14}+\frac{1424314263809}{578649435328}a^{13}-\frac{10898906902719}{578649435328}a^{12}+\frac{6343032261785}{578649435328}a^{11}+\frac{15341905437881}{578649435328}a^{10}-\frac{18913369187827}{578649435328}a^{9}-\frac{14048401201219}{578649435328}a^{8}+\frac{23118287100389}{578649435328}a^{7}+\frac{6319577821861}{578649435328}a^{6}-\frac{16834079217695}{578649435328}a^{5}-\frac{262813125261}{289324717664}a^{4}+\frac{9502578675477}{1157298870656}a^{3}-\frac{2070075445679}{1157298870656}a^{2}+\frac{5355361971721}{1157298870656}a-\frac{3708346015125}{1157298870656}$, $\frac{16089452013}{578649435328}a^{20}-\frac{17548746351}{289324717664}a^{19}-\frac{21856414013}{144662358832}a^{18}+\frac{169017638937}{578649435328}a^{17}-\frac{5242378001}{144662358832}a^{16}-\frac{247523631651}{289324717664}a^{15}+\frac{194248956357}{289324717664}a^{14}+\frac{14750844585}{18082794854}a^{13}-\frac{27158031835}{9041397427}a^{12}+\frac{58938698615}{289324717664}a^{11}+\frac{1391455292931}{289324717664}a^{10}-\frac{435926288707}{144662358832}a^{9}-\frac{915434320323}{144662358832}a^{8}+\frac{1187941773949}{289324717664}a^{7}+\frac{979254130849}{289324717664}a^{6}-\frac{374257690087}{72331179416}a^{5}-\frac{884382196413}{578649435328}a^{4}+\frac{376067881649}{144662358832}a^{3}-\frac{161403691203}{289324717664}a^{2}-\frac{661133074917}{578649435328}a+\frac{4901810997}{9041397427}$, $\frac{596854485}{578649435328}a^{20}+\frac{54738701309}{1157298870656}a^{19}-\frac{86242213247}{1157298870656}a^{18}-\frac{372447403255}{1157298870656}a^{17}+\frac{284018345067}{1157298870656}a^{16}+\frac{107102601553}{578649435328}a^{15}-\frac{672805223615}{578649435328}a^{14}+\frac{7284441729}{578649435328}a^{13}+\frac{799000775169}{578649435328}a^{12}-\frac{1809294548679}{578649435328}a^{11}-\frac{1664278463639}{578649435328}a^{10}+\frac{3411423357325}{578649435328}a^{9}+\frac{1236688604861}{578649435328}a^{8}-\frac{5569920789019}{578649435328}a^{7}-\frac{2939011913259}{578649435328}a^{6}+\frac{2985599791105}{578649435328}a^{5}+\frac{121812064719}{144662358832}a^{4}-\frac{3024670686115}{1157298870656}a^{3}-\frac{515121661671}{1157298870656}a^{2}-\frac{1938633392639}{1157298870656}a+\frac{698342925247}{1157298870656}$, $\frac{175280861911}{1157298870656}a^{20}-\frac{474345335353}{1157298870656}a^{19}-\frac{842727929861}{1157298870656}a^{18}+\frac{2545962503109}{1157298870656}a^{17}-\frac{200299827153}{289324717664}a^{16}-\frac{2963218437203}{578649435328}a^{15}+\frac{3676671911565}{578649435328}a^{14}+\frac{2075541487619}{578649435328}a^{13}-\frac{11162835527285}{578649435328}a^{12}+\frac{5290248244545}{578649435328}a^{11}+\frac{17010187305589}{578649435328}a^{10}-\frac{17399443180533}{578649435328}a^{9}-\frac{16924064124733}{578649435328}a^{8}+\frac{21301089653217}{578649435328}a^{7}+\frac{9141061001485}{578649435328}a^{6}-\frac{14444410115453}{578649435328}a^{5}-\frac{3611798241913}{1157298870656}a^{4}+\frac{6744952422995}{1157298870656}a^{3}-\frac{2045579758673}{1157298870656}a^{2}+\frac{5565662366129}{1157298870656}a-\frac{1770253298615}{578649435328}$, $\frac{150829193581}{1157298870656}a^{20}-\frac{27030032205}{72331179416}a^{19}-\frac{155989835767}{289324717664}a^{18}+\frac{1113139812617}{578649435328}a^{17}-\frac{1213518456829}{1157298870656}a^{16}-\frac{1158648299527}{289324717664}a^{15}+\frac{1750259688207}{289324717664}a^{14}+\frac{54258396817}{36165589708}a^{13}-\frac{2367945883271}{144662358832}a^{12}+\frac{3143193015583}{289324717664}a^{11}+\frac{6217762521895}{289324717664}a^{10}-\frac{4227533373839}{144662358832}a^{9}-\frac{1268707592147}{72331179416}a^{8}+\frac{9675108434837}{289324717664}a^{7}+\frac{1317275909569}{289324717664}a^{6}-\frac{815099808675}{36165589708}a^{5}+\frac{3364234441443}{1157298870656}a^{4}+\frac{1248927194331}{289324717664}a^{3}-\frac{79071756327}{36165589708}a^{2}+\frac{2966272951331}{578649435328}a-\frac{4418522544091}{1157298870656}$, $\frac{3359282503}{144662358832}a^{20}-\frac{66654576101}{1157298870656}a^{19}-\frac{151532555141}{1157298870656}a^{18}+\frac{338261364355}{1157298870656}a^{17}+\frac{87861959335}{1157298870656}a^{16}-\frac{426144093597}{578649435328}a^{15}+\frac{276532168247}{578649435328}a^{14}+\frac{649253579135}{578649435328}a^{13}-\frac{1331297534653}{578649435328}a^{12}-\frac{59851945197}{578649435328}a^{11}+\frac{2957563553807}{578649435328}a^{10}-\frac{638342218053}{578649435328}a^{9}-\frac{4048852935457}{578649435328}a^{8}+\frac{1024368592895}{578649435328}a^{7}+\frac{4303583226683}{578649435328}a^{6}+\frac{581903411775}{578649435328}a^{5}-\frac{2196665477569}{578649435328}a^{4}-\frac{2259113796893}{1157298870656}a^{3}+\frac{3071011350219}{1157298870656}a^{2}+\frac{2215155328163}{1157298870656}a+\frac{87945479831}{1157298870656}$, $\frac{48263755047}{578649435328}a^{20}-\frac{263242917151}{1157298870656}a^{19}-\frac{442129772493}{1157298870656}a^{18}+\frac{1402219910317}{1157298870656}a^{17}-\frac{623484547327}{1157298870656}a^{16}-\frac{1587754015015}{578649435328}a^{15}+\frac{2137511821311}{578649435328}a^{14}+\frac{764403052657}{578649435328}a^{13}-\frac{6092355161441}{578649435328}a^{12}+\frac{3401115447985}{578649435328}a^{11}+\frac{8398381718951}{578649435328}a^{10}-\frac{10321679352147}{578649435328}a^{9}-\frac{7516351947837}{578649435328}a^{8}+\frac{11460785465485}{578649435328}a^{7}+\frac{3257201988475}{578649435328}a^{6}-\frac{7872140844655}{578649435328}a^{5}-\frac{123910516381}{72331179416}a^{4}+\frac{1704601063209}{1157298870656}a^{3}+\frac{263322172723}{1157298870656}a^{2}+\frac{4197240082333}{1157298870656}a-\frac{2685712297355}{1157298870656}$, $\frac{39316738907}{289324717664}a^{20}-\frac{209022228451}{578649435328}a^{19}-\frac{390454918837}{578649435328}a^{18}+\frac{572530078401}{289324717664}a^{17}-\frac{162322627653}{289324717664}a^{16}-\frac{698664710397}{144662358832}a^{15}+\frac{828641734339}{144662358832}a^{14}+\frac{1066380406143}{289324717664}a^{13}-\frac{5231200325489}{289324717664}a^{12}+\frac{542450349773}{72331179416}a^{11}+\frac{2054130684897}{72331179416}a^{10}-\frac{8204460044927}{289324717664}a^{9}-\frac{8504501869023}{289324717664}a^{8}+\frac{5423249163725}{144662358832}a^{7}+\frac{2273792544077}{144662358832}a^{6}-\frac{8858615353949}{289324717664}a^{5}-\frac{67969170667}{36165589708}a^{4}+\frac{7018655491867}{578649435328}a^{3}-\frac{2668113096115}{578649435328}a^{2}+\frac{10269039948}{9041397427}a-\frac{61247657997}{144662358832}$, $\frac{55984860469}{1157298870656}a^{20}-\frac{8847417263}{72331179416}a^{19}-\frac{78366406115}{289324717664}a^{18}+\frac{94586205003}{144662358832}a^{17}+\frac{94267975449}{1157298870656}a^{16}-\frac{58879832119}{36165589708}a^{15}+\frac{47495410085}{36165589708}a^{14}+\frac{296535077493}{144662358832}a^{13}-\frac{767776510765}{144662358832}a^{12}+\frac{3111103409}{9041397427}a^{11}+\frac{1549189302641}{144662358832}a^{10}-\frac{649646319205}{144662358832}a^{9}-\frac{2116162921503}{144662358832}a^{8}+\frac{436543789581}{72331179416}a^{7}+\frac{1933482851545}{144662358832}a^{6}-\frac{429233354709}{144662358832}a^{5}-\frac{6966646160741}{1157298870656}a^{4}-\frac{1472824249}{18082794854}a^{3}+\frac{79756044051}{289324717664}a^{2}+\frac{127959377307}{72331179416}a-\frac{145589222425}{1157298870656}$, $\frac{99512973441}{1157298870656}a^{20}-\frac{302722324513}{1157298870656}a^{19}-\frac{376806790615}{1157298870656}a^{18}+\frac{1606020582627}{1157298870656}a^{17}-\frac{507199169487}{578649435328}a^{16}-\frac{1648988284889}{578649435328}a^{15}+\frac{2636552399185}{578649435328}a^{14}+\frac{472755449139}{578649435328}a^{13}-\frac{6877599018147}{578649435328}a^{12}+\frac{4981376767183}{578649435328}a^{11}+\frac{8757130585205}{578649435328}a^{10}-\frac{13600648825681}{578649435328}a^{9}-\frac{6961348706863}{578649435328}a^{8}+\frac{16452205934651}{578649435328}a^{7}+\frac{2221781600529}{578649435328}a^{6}-\frac{12097830568133}{578649435328}a^{5}+\frac{20480587745}{1157298870656}a^{4}+\frac{5765033739335}{1157298870656}a^{3}-\frac{893216980847}{1157298870656}a^{2}+\frac{3482133593755}{1157298870656}a-\frac{426409143305}{144662358832}$ 129427364769/1157298870656a^20 - 158973451327/578649435328a^19 - 351863426395/578649435328a^18 + 213741308075/144662358832a^17 - 147170050183/1157298870656a^16 - 139141792923/36165589708a^15 + 133303594185/36165589708a^14 + 1065014444933/289324717664a^13 - 3862091276047/289324717664a^12 + 448956981469/144662358832a^11 + 1651205609303/72331179416a^10 - 4704181620609/289324717664a^9 - 7740895406949/289324717664a^8 + 750934815771/36165589708a^7 + 2688136179979/144662358832a^6 - 4202981530531/289324717664a^5 - 8711131285301/1157298870656a^4 + 2051237676699/578649435328a^3 + 375446281071/578649435328a^2 + 1194438179365/289324717664a - 748174762069/1157298870656, 83285702587/578649435328a^20 - 471299973371/1157298870656a^19 - 731066570167/1157298870656a^18 + 2508421152449/1157298870656a^17 - 1162863783073/1157298870656a^16 - 2769265833775/578649435328a^15 + 3859091009665/578649435328a^14 + 1424314263809/578649435328a^13 - 10898906902719/578649435328a^12 + 6343032261785/578649435328a^11 + 15341905437881/578649435328a^10 - 18913369187827/578649435328a^9 - 14048401201219/578649435328a^8 + 23118287100389/578649435328a^7 + 6319577821861/578649435328a^6 - 16834079217695/578649435328a^5 - 262813125261/289324717664a^4 + 9502578675477/1157298870656a^3 - 2070075445679/1157298870656a^2 + 5355361971721/1157298870656a - 3708346015125/1157298870656, 16089452013/578649435328a^20 - 17548746351/289324717664a^19 - 21856414013/144662358832a^18 + 169017638937/578649435328a^17 - 5242378001/144662358832a^16 - 247523631651/289324717664a^15 + 194248956357/289324717664a^14 + 14750844585/18082794854a^13 - 27158031835/9041397427a^12 + 58938698615/289324717664a^11 + 1391455292931/289324717664a^10 - 435926288707/144662358832a^9 - 915434320323/144662358832a^8 + 1187941773949/289324717664a^7 + 979254130849/289324717664a^6 - 374257690087/72331179416a^5 - 884382196413/578649435328a^4 + 376067881649/144662358832a^3 - 161403691203/289324717664a^2 - 661133074917/578649435328a + 4901810997/9041397427, 596854485/578649435328a^20 + 54738701309/1157298870656a^19 - 86242213247/1157298870656a^18 - 372447403255/1157298870656a^17 + 284018345067/1157298870656a^16 + 107102601553/578649435328a^15 - 672805223615/578649435328a^14 + 7284441729/578649435328a^13 + 799000775169/578649435328a^12 - 1809294548679/578649435328a^11 - 1664278463639/578649435328a^10 + 3411423357325/578649435328a^9 + 1236688604861/578649435328a^8 - 5569920789019/578649435328a^7 - 2939011913259/578649435328a^6 + 2985599791105/578649435328a^5 + 121812064719/144662358832a^4 - 3024670686115/1157298870656a^3 - 515121661671/1157298870656a^2 - 1938633392639/1157298870656a + 698342925247/1157298870656, 175280861911/1157298870656a^20 - 474345335353/1157298870656a^19 - 842727929861/1157298870656a^18 + 2545962503109/1157298870656a^17 - 200299827153/289324717664a^16 - 2963218437203/578649435328a^15 + 3676671911565/578649435328a^14 + 2075541487619/578649435328a^13 - 11162835527285/578649435328a^12 + 5290248244545/578649435328a^11 + 17010187305589/578649435328a^10 - 17399443180533/578649435328a^9 - 16924064124733/578649435328a^8 + 21301089653217/578649435328a^7 + 9141061001485/578649435328a^6 - 14444410115453/578649435328a^5 - 3611798241913/1157298870656a^4 + 6744952422995/1157298870656a^3 - 2045579758673/1157298870656a^2 + 5565662366129/1157298870656a - 1770253298615/578649435328, 150829193581/1157298870656a^20 - 27030032205/72331179416a^19 - 155989835767/289324717664a^18 + 1113139812617/578649435328a^17 - 1213518456829/1157298870656a^16 - 1158648299527/289324717664a^15 + 1750259688207/289324717664a^14 + 54258396817/36165589708a^13 - 2367945883271/144662358832a^12 + 3143193015583/289324717664a^11 + 6217762521895/289324717664a^10 - 4227533373839/144662358832a^9 - 1268707592147/72331179416a^8 + 9675108434837/289324717664a^7 + 1317275909569/289324717664a^6 - 815099808675/36165589708a^5 + 3364234441443/1157298870656a^4 + 1248927194331/289324717664a^3 - 79071756327/36165589708a^2 + 2966272951331/578649435328a - 4418522544091/1157298870656, 3359282503/144662358832a^20 - 66654576101/1157298870656a^19 - 151532555141/1157298870656a^18 + 338261364355/1157298870656a^17 + 87861959335/1157298870656a^16 - 426144093597/578649435328a^15 + 276532168247/578649435328a^14 + 649253579135/578649435328a^13 - 1331297534653/578649435328a^12 - 59851945197/578649435328a^11 + 2957563553807/578649435328a^10 - 638342218053/578649435328a^9 - 4048852935457/578649435328a^8 + 1024368592895/578649435328a^7 + 4303583226683/578649435328a^6 + 581903411775/578649435328a^5 - 2196665477569/578649435328a^4 - 2259113796893/1157298870656a^3 + 3071011350219/1157298870656a^2 + 2215155328163/1157298870656a + 87945479831/1157298870656, 48263755047/578649435328a^20 - 263242917151/1157298870656a^19 - 442129772493/1157298870656a^18 + 1402219910317/1157298870656a^17 - 623484547327/1157298870656a^16 - 1587754015015/578649435328a^15 + 2137511821311/578649435328a^14 + 764403052657/578649435328a^13 - 6092355161441/578649435328a^12 + 3401115447985/578649435328a^11 + 8398381718951/578649435328a^10 - 10321679352147/578649435328a^9 - 7516351947837/578649435328a^8 + 11460785465485/578649435328a^7 + 3257201988475/578649435328a^6 - 7872140844655/578649435328a^5 - 123910516381/72331179416a^4 + 1704601063209/1157298870656a^3 + 263322172723/1157298870656a^2 + 4197240082333/1157298870656a - 2685712297355/1157298870656, 39316738907/289324717664a^20 - 209022228451/578649435328a^19 - 390454918837/578649435328a^18 + 572530078401/289324717664a^17 - 162322627653/289324717664a^16 - 698664710397/144662358832a^15 + 828641734339/144662358832a^14 + 1066380406143/289324717664a^13 - 5231200325489/289324717664a^12 + 542450349773/72331179416a^11 + 2054130684897/72331179416a^10 - 8204460044927/289324717664a^9 - 8504501869023/289324717664a^8 + 5423249163725/144662358832a^7 + 2273792544077/144662358832a^6 - 8858615353949/289324717664a^5 - 67969170667/36165589708a^4 + 7018655491867/578649435328a^3 - 2668113096115/578649435328a^2 + 10269039948/9041397427a - 61247657997/144662358832, 55984860469/1157298870656a^20 - 8847417263/72331179416a^19 - 78366406115/289324717664a^18 + 94586205003/144662358832a^17 + 94267975449/1157298870656a^16 - 58879832119/36165589708a^15 + 47495410085/36165589708a^14 + 296535077493/144662358832a^13 - 767776510765/144662358832a^12 + 3111103409/9041397427a^11 + 1549189302641/144662358832a^10 - 649646319205/144662358832a^9 - 2116162921503/144662358832a^8 + 436543789581/72331179416a^7 + 1933482851545/144662358832a^6 - 429233354709/144662358832a^5 - 6966646160741/1157298870656a^4 - 1472824249/18082794854a^3 + 79756044051/289324717664a^2 + 127959377307/72331179416a - 145589222425/1157298870656, 99512973441/1157298870656a^20 - 302722324513/1157298870656a^19 - 376806790615/1157298870656a^18 + 1606020582627/1157298870656a^17 - 507199169487/578649435328a^16 - 1648988284889/578649435328a^15 + 2636552399185/578649435328a^14 + 472755449139/578649435328a^13 - 6877599018147/578649435328a^12 + 4981376767183/578649435328a^11 + 8757130585205/578649435328a^10 - 13600648825681/578649435328a^9 - 6961348706863/578649435328a^8 + 16452205934651/578649435328a^7 + 2221781600529/578649435328a^6 - 12097830568133/578649435328a^5 + 20480587745/1157298870656a^4 + 5765033739335/1157298870656a^3 - 893216980847/1157298870656a^2 + 3482133593755/1157298870656a - 426409143305/144662358832 (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:	$ 3668230.07566 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 3668230.07566 \cdot 1}{2\cdot\sqrt{7146646609494406531041460224}}\cr\approx \mathstrut & 2.64901379622 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*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^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*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^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*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^21 - 3*x^20 - 4*x^19 + 16*x^18 - 9*x^17 - 33*x^16 + 52*x^15 + 12*x^14 - 136*x^13 + 96*x^12 + 180*x^11 - 260*x^10 - 144*x^9 + 312*x^8 + 44*x^7 - 220*x^6 + 15*x^5 + 59*x^4 - 16*x^3 + 36*x^2 - 31*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

$\PGL(2,7)$ (as 21T20):

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 non-solvable group of order 336

The 9 conjugacy class representatives for $\PGL(2,7)$

Character table for $\PGL(2,7)$

Intermediate fields

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

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 8 sibling:	8.2.195689447424.8
Degree 14 sibling:	data not computed
Degree 16 sibling:	data not computed
Degree 24 sibling:	data not computed
Degree 28 siblings:	data not computed
Degree 42 siblings:	data not computed
Minimal sibling:	8.2.195689447424.8

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	${\href{/padicField/5.7.0.1}{7} }^{3}$	${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.3.0.1}{3} }^{7}$	${\href{/padicField/17.3.0.1}{3} }^{7}$	${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$	${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.3.0.1}{3} }^{7}$	${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.7.0.1}{7} }^{3}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.4.9.3	$x^{4} + 4 x^{3} + 2 x^{2} + 2$	$4$	$1$	$9$	$D_{4}$	$[2, 3, 7/2]$
	2.8.28.65	$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 16 x^{2} + 10$	$8$	$1$	$28$	$D_{8}$	$[2, 3, 7/2, 9/2]$
	2.8.27.41	$x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{2} + 2$	$8$	$1$	$27$	$D_{8}$	$[2, 3, 7/2, 9/2]$
$3$ 3		Deg $21$	$7$	$3$	$18$

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Elliptic curves over $\Q$
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