LMFDB - Number field 20.0.1597098119556903839111328125.2

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331)

gp:K = bnfinit(y^20 - 12*y^15 + 179*y^10 - 858*y^5 + 1331, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331)

$ x^{20} - 12x^{15} + 179x^{10} - 858x^{5} + 1331 $ x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$20$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(0, 10)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$1597098119556903839111328125$ 1597098119556903839111328125 $\medspace = 5^{27}\cdot 11^{8}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$22.92$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$5^{143/100}11^{4/5}\approx 68.02072465087673$
Ramified primes:	$5$, $11$ 5, 11	comment:Ramified primes 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}) $
$\Aut(K/\Q)$:	$C_5$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:	$\Q(\zeta_{5})$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{9}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{110}a^{10}+\frac{49}{110}a^{5}-\frac{1}{10}$, $\frac{1}{110}a^{11}+\frac{49}{110}a^{6}-\frac{1}{10}a$, $\frac{1}{110}a^{12}+\frac{1}{22}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{10}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{110}a^{13}+\frac{1}{22}a^{8}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{10}a^{3}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{550}a^{14}-\frac{1}{275}a^{13}-\frac{1}{550}a^{12}+\frac{1}{275}a^{11}+\frac{1}{550}a^{10}+\frac{49}{550}a^{9}+\frac{6}{275}a^{8}-\frac{49}{550}a^{7}-\frac{61}{275}a^{6}-\frac{61}{550}a^{5}-\frac{21}{50}a^{4}+\frac{6}{25}a^{3}+\frac{21}{50}a^{2}+\frac{9}{25}a+\frac{9}{50}$, $\frac{1}{550}a^{15}+\frac{1}{550}a^{10}+\frac{167}{550}a^{5}+\frac{4}{25}$, $\frac{1}{550}a^{16}+\frac{1}{550}a^{11}+\frac{167}{550}a^{6}+\frac{4}{25}a$, $\frac{1}{550}a^{17}+\frac{1}{550}a^{12}-\frac{53}{550}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{9}{25}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{6050}a^{18}+\frac{21}{6050}a^{13}-\frac{503}{6050}a^{8}-\frac{6}{275}a^{3}$, $\frac{1}{6050}a^{19}-\frac{1}{6050}a^{14}-\frac{1}{550}a^{13}+\frac{1}{275}a^{12}+\frac{1}{550}a^{11}-\frac{1}{275}a^{10}-\frac{371}{6050}a^{9}-\frac{49}{550}a^{8}-\frac{6}{275}a^{7}-\frac{171}{550}a^{6}-\frac{104}{275}a^{5}+\frac{12}{55}a^{4}+\frac{21}{50}a^{3}-\frac{6}{25}a^{2}-\frac{11}{50}a+\frac{11}{25}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/5*a^7 - 1/5*a^6 - 1/5*a^5 + 2/5*a^2 - 2/5*a - 2/5, 1/5*a^8 - 2/5*a^6 - 1/5*a^5 + 2/5*a^3 + 1/5*a - 2/5, 1/5*a^9 + 2/5*a^6 - 2/5*a^5 + 2/5*a^4 - 1/5*a + 1/5, 1/110*a^10 + 49/110*a^5 - 1/10, 1/110*a^11 + 49/110*a^6 - 1/10*a, 1/110*a^12 + 1/22*a^7 + 2/5*a^6 + 2/5*a^5 + 1/10*a^2 - 1/5*a - 1/5, 1/110*a^13 + 1/22*a^8 - 1/5*a^6 + 2/5*a^5 + 1/10*a^3 - 2/5*a - 1/5, 1/550*a^14 - 1/275*a^13 - 1/550*a^12 + 1/275*a^11 + 1/550*a^10 + 49/550*a^9 + 6/275*a^8 - 49/550*a^7 - 61/275*a^6 - 61/550*a^5 - 21/50*a^4 + 6/25*a^3 + 21/50*a^2 + 9/25*a + 9/50, 1/550*a^15 + 1/550*a^10 + 167/550*a^5 + 4/25, 1/550*a^16 + 1/550*a^11 + 167/550*a^6 + 4/25*a, 1/550*a^17 + 1/550*a^12 - 53/550*a^7 + 2/5*a^6 + 2/5*a^5 + 9/25*a^2 - 1/5*a - 1/5, 1/6050*a^18 + 21/6050*a^13 - 503/6050*a^8 - 6/275*a^3, 1/6050*a^19 - 1/6050*a^14 - 1/550*a^13 + 1/275*a^12 + 1/550*a^11 - 1/275*a^10 - 371/6050*a^9 - 49/550*a^8 - 6/275*a^7 - 171/550*a^6 - 104/275*a^5 + 12/55*a^4 + 21/50*a^3 - 6/25*a^2 - 11/50*a + 11/25

comment:Integral basis

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

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$9$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ \frac{3}{550} a^{15} - \frac{1}{25} a^{10} + \frac{188}{275} a^{5} - \frac{51}{50} $ (3)/(550)a^(15) - (1)/(25)a^(10) + (188)/(275)*a^(5) - (51)/(50) (order $10$)	comment:Generator for roots of unity 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:	$\frac{1}{550}a^{15}-\frac{9}{550}a^{10}+\frac{227}{550}a^{5}-\frac{41}{25}$, $\frac{19}{6050}a^{19}+\frac{13}{6050}a^{18}+\frac{2}{275}a^{16}-\frac{1}{550}a^{15}-\frac{92}{3025}a^{14}-\frac{101}{6050}a^{13}-\frac{1}{275}a^{12}-\frac{37}{550}a^{11}+\frac{1}{50}a^{10}+\frac{1508}{3025}a^{9}+\frac{351}{1210}a^{8}+\frac{6}{275}a^{7}+\frac{639}{550}a^{6}-\frac{239}{550}a^{5}-\frac{171}{110}a^{4}-\frac{56}{275}a^{3}-\frac{19}{25}a^{2}-\frac{147}{50}a+\frac{7}{5}$, $\frac{19}{6050}a^{19}+\frac{4}{3025}a^{18}+\frac{3}{550}a^{17}+\frac{1}{275}a^{16}+\frac{1}{275}a^{15}-\frac{92}{3025}a^{14}-\frac{48}{3025}a^{13}-\frac{29}{550}a^{12}-\frac{19}{550}a^{11}-\frac{13}{275}a^{10}+\frac{1508}{3025}a^{9}+\frac{119}{605}a^{8}+\frac{43}{50}a^{7}+\frac{59}{110}a^{6}+\frac{196}{275}a^{5}-\frac{171}{110}a^{4}-\frac{301}{275}a^{3}-\frac{62}{25}a^{2}-\frac{113}{50}a-\frac{68}{25}$, $\frac{7}{3025}a^{19}-\frac{3}{3025}a^{18}-\frac{1}{550}a^{17}-\frac{1}{550}a^{16}+\frac{3}{550}a^{15}-\frac{27}{1210}a^{14}+\frac{17}{6050}a^{13}+\frac{4}{275}a^{12}+\frac{8}{275}a^{11}-\frac{8}{275}a^{10}+\frac{2187}{6050}a^{9}-\frac{173}{1210}a^{8}-\frac{83}{275}a^{7}-\frac{107}{275}a^{6}+\frac{9}{11}a^{5}-\frac{509}{550}a^{4}+\frac{39}{550}a^{3}+\frac{23}{50}a^{2}+\frac{3}{2}a-\frac{67}{50}$, $\frac{19}{6050}a^{19}-\frac{23}{6050}a^{18}+\frac{1}{275}a^{17}-\frac{3}{550}a^{16}-\frac{92}{3025}a^{14}+\frac{243}{6050}a^{13}-\frac{3}{110}a^{12}+\frac{21}{550}a^{11}-\frac{3}{550}a^{10}+\frac{1508}{3025}a^{9}-\frac{3677}{6050}a^{8}+\frac{271}{550}a^{7}-\frac{17}{22}a^{6}+\frac{73}{550}a^{5}-\frac{171}{110}a^{4}+\frac{109}{55}a^{3}-\frac{37}{50}a^{2}+\frac{36}{25}a+\frac{3}{50}$, $\frac{2}{3025}a^{19}-\frac{19}{6050}a^{18}-\frac{1}{275}a^{17}-\frac{1}{275}a^{16}-\frac{1}{550}a^{15}-\frac{3}{1210}a^{14}+\frac{92}{3025}a^{13}+\frac{17}{550}a^{12}+\frac{1}{22}a^{11}+\frac{2}{55}a^{10}+\frac{397}{6050}a^{9}-\frac{1508}{3025}a^{8}-\frac{283}{550}a^{7}-\frac{331}{550}a^{6}-\frac{119}{275}a^{5}+\frac{51}{550}a^{4}+\frac{171}{110}a^{3}+\frac{3}{2}a^{2}+\frac{127}{50}a+\frac{101}{50}$, $\frac{1}{1210}a^{19}+\frac{9}{6050}a^{18}-\frac{1}{550}a^{17}-\frac{1}{275}a^{16}+\frac{2}{275}a^{15}-\frac{49}{6050}a^{14}-\frac{54}{3025}a^{13}+\frac{4}{275}a^{12}+\frac{1}{22}a^{11}-\frac{4}{55}a^{10}+\frac{829}{6050}a^{9}+\frac{149}{605}a^{8}-\frac{83}{275}a^{7}-\frac{331}{550}a^{6}+\frac{301}{275}a^{5}-\frac{173}{275}a^{4}-\frac{581}{550}a^{3}+\frac{23}{50}a^{2}+\frac{127}{50}a-\frac{57}{25}$, $\frac{23}{6050}a^{19}-\frac{3}{1210}a^{18}-\frac{1}{550}a^{17}+\frac{2}{275}a^{16}-\frac{4}{275}a^{15}-\frac{199}{6050}a^{14}+\frac{169}{6050}a^{13}+\frac{3}{275}a^{12}-\frac{3}{55}a^{11}+\frac{7}{55}a^{10}+\frac{3413}{6050}a^{9}-\frac{2619}{6050}a^{8}-\frac{7}{25}a^{7}+\frac{271}{275}a^{6}-\frac{57}{25}a^{5}-\frac{402}{275}a^{4}+\frac{453}{275}a^{3}-\frac{3}{10}a^{2}-\frac{37}{25}a+\frac{134}{25}$, $\frac{9}{3025}a^{18}-\frac{3}{550}a^{17}+\frac{1}{275}a^{16}+\frac{1}{275}a^{15}-\frac{86}{3025}a^{13}+\frac{27}{550}a^{12}-\frac{9}{275}a^{11}-\frac{13}{550}a^{10}+\frac{1358}{3025}a^{8}-\frac{461}{550}a^{7}+\frac{172}{275}a^{6}+\frac{259}{550}a^{5}-\frac{438}{275}a^{3}+\frac{68}{25}a^{2}-\frac{42}{25}a-\frac{49}{50}$ 1/550a^15 - 9/550a^10 + 227/550a^5 - 41/25, 19/6050a^19 + 13/6050a^18 + 2/275a^16 - 1/550a^15 - 92/3025a^14 - 101/6050a^13 - 1/275a^12 - 37/550a^11 + 1/50a^10 + 1508/3025a^9 + 351/1210a^8 + 6/275a^7 + 639/550a^6 - 239/550a^5 - 171/110a^4 - 56/275a^3 - 19/25a^2 - 147/50a + 7/5, 19/6050a^19 + 4/3025a^18 + 3/550a^17 + 1/275a^16 + 1/275a^15 - 92/3025a^14 - 48/3025a^13 - 29/550a^12 - 19/550a^11 - 13/275a^10 + 1508/3025a^9 + 119/605a^8 + 43/50a^7 + 59/110a^6 + 196/275a^5 - 171/110a^4 - 301/275a^3 - 62/25a^2 - 113/50a - 68/25, 7/3025a^19 - 3/3025a^18 - 1/550a^17 - 1/550a^16 + 3/550a^15 - 27/1210a^14 + 17/6050a^13 + 4/275a^12 + 8/275a^11 - 8/275a^10 + 2187/6050a^9 - 173/1210a^8 - 83/275a^7 - 107/275a^6 + 9/11a^5 - 509/550a^4 + 39/550a^3 + 23/50a^2 + 3/2a - 67/50, 19/6050a^19 - 23/6050a^18 + 1/275a^17 - 3/550a^16 - 92/3025a^14 + 243/6050a^13 - 3/110a^12 + 21/550a^11 - 3/550a^10 + 1508/3025a^9 - 3677/6050a^8 + 271/550a^7 - 17/22a^6 + 73/550a^5 - 171/110a^4 + 109/55a^3 - 37/50a^2 + 36/25a + 3/50, 2/3025a^19 - 19/6050a^18 - 1/275a^17 - 1/275a^16 - 1/550a^15 - 3/1210a^14 + 92/3025a^13 + 17/550a^12 + 1/22a^11 + 2/55a^10 + 397/6050a^9 - 1508/3025a^8 - 283/550a^7 - 331/550a^6 - 119/275a^5 + 51/550a^4 + 171/110a^3 + 3/2a^2 + 127/50a + 101/50, 1/1210a^19 + 9/6050a^18 - 1/550a^17 - 1/275a^16 + 2/275a^15 - 49/6050a^14 - 54/3025a^13 + 4/275a^12 + 1/22a^11 - 4/55a^10 + 829/6050a^9 + 149/605a^8 - 83/275a^7 - 331/550a^6 + 301/275a^5 - 173/275a^4 - 581/550a^3 + 23/50a^2 + 127/50a - 57/25, 23/6050a^19 - 3/1210a^18 - 1/550a^17 + 2/275a^16 - 4/275a^15 - 199/6050a^14 + 169/6050a^13 + 3/275a^12 - 3/55a^11 + 7/55a^10 + 3413/6050a^9 - 2619/6050a^8 - 7/25a^7 + 271/275a^6 - 57/25a^5 - 402/275a^4 + 453/275a^3 - 3/10a^2 - 37/25a + 134/25, 9/3025a^18 - 3/550a^17 + 1/275a^16 + 1/275a^15 - 86/3025a^13 + 27/550a^12 - 9/275a^11 - 13/550a^10 + 1358/3025a^8 - 461/550a^7 + 172/275a^6 + 259/550a^5 - 438/275a^3 + 68/25a^2 - 42/25*a - 49/50	comment:Fundamental units 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:	$ 1387068.91153 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1387068.91153 \cdot 1}{10\cdot\sqrt{1597098119556903839111328125}}\cr\approx \mathstrut & 0.33283648154 \end{aligned}\]

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 12*x^15 + 179*x^10 - 858*x^5 + 1331); 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

$C_5^3:C_4$ (as 20T125):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 500

The 38 conjugacy class representatives for $C_5^3:C_4$

Character table for $C_5^3:C_4$

Intermediate fields

$\Q(\sqrt{5}) $, $\Q(\zeta_{5})$

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 20 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	${\href{/padicField/2.4.0.1}{4} }^{5}$	${\href{/padicField/3.4.0.1}{4} }^{5}$	R	${\href{/padicField/7.4.0.1}{4} }^{5}$	R	${\href{/padicField/13.4.0.1}{4} }^{5}$	${\href{/padicField/17.4.0.1}{4} }^{5}$	${\href{/padicField/19.10.0.1}{10} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{5}$	${\href{/padicField/29.10.0.1}{10} }^{2}$	${\href{/padicField/31.5.0.1}{5} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.5.0.1}{5} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{5}$	${\href{/padicField/47.4.0.1}{4} }^{5}$	${\href{/padicField/53.4.0.1}{4} }^{5}$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
$5$ 5	5.1.20.27a1.4	$x^{20} + 5 x^{9} + 15 x^{8} + 5$	$20$	$1$	$27$	20T26	not computed
$11$ 11	11.1.5.4a1.1	$x^{5} + 11$	$5$	$1$	$4$	$C_5$	$$[\ ]_{5}$$
	11.1.5.4a1.3	$x^{5} + 44$	$5$	$1$	$4$	$C_5$	$$[\ ]_{5}$$
	11.5.1.0a1.1	$x^{5} + 10 x^{2} + 9$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$
	11.5.1.0a1.1	$x^{5} + 10 x^{2} + 9$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$

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