LMFDB - Number field 20.0.23383113568432629108428955078125.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561)

gp:K = bnfinit(y^20 - 36*y^15 + 2486*y^10 - 47916*y^5 + 1771561, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561)

$ x^{20} - 36x^{15} + 2486x^{10} - 47916x^{5} + 1771561 $ x^20 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561

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:	$23383113568432629108428955078125$ 23383113568432629108428955078125 $\medspace = 5^{27}\cdot 11^{12}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$37.02$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{10}a^{7}+\frac{1}{5}a^{6}+\frac{1}{10}a^{5}+\frac{1}{10}a^{2}+\frac{1}{5}a+\frac{1}{10}$, $\frac{1}{10}a^{8}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{10}a^{3}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{10}a^{9}-\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{1}{10}a^{4}-\frac{1}{10}a-\frac{1}{5}$, $\frac{1}{660}a^{10}+\frac{27}{110}a^{5}+\frac{11}{60}$, $\frac{1}{660}a^{11}+\frac{27}{110}a^{6}+\frac{11}{60}a$, $\frac{1}{660}a^{12}+\frac{1}{22}a^{7}+\frac{1}{10}a^{6}-\frac{1}{5}a^{5}-\frac{1}{60}a^{2}+\frac{1}{10}a-\frac{1}{5}$, $\frac{1}{7260}a^{13}-\frac{3}{605}a^{8}-\frac{1}{10}a^{6}+\frac{1}{10}a^{5}-\frac{247}{660}a^{3}-\frac{1}{10}a+\frac{1}{10}$, $\frac{1}{36300}a^{14}-\frac{1}{36300}a^{13}+\frac{1}{3300}a^{12}-\frac{1}{3300}a^{11}+\frac{1}{3300}a^{10}-\frac{124}{3025}a^{9}+\frac{124}{3025}a^{8}+\frac{27}{550}a^{7}-\frac{27}{550}a^{6}+\frac{27}{550}a^{5}+\frac{1601}{3300}a^{4}-\frac{1601}{3300}a^{3}-\frac{49}{300}a^{2}+\frac{49}{300}a-\frac{49}{300}$, $\frac{1}{798600}a^{15}-\frac{157}{798600}a^{10}-\frac{10147}{72600}a^{5}-\frac{89}{600}$, $\frac{1}{798600}a^{16}-\frac{157}{798600}a^{11}-\frac{10147}{72600}a^{6}-\frac{89}{600}a$, $\frac{1}{8784600}a^{17}-\frac{1367}{8784600}a^{12}-\frac{20707}{798600}a^{7}+\frac{1}{5}a^{6}+\frac{1}{10}a^{5}-\frac{1939}{6600}a^{2}+\frac{1}{5}a+\frac{1}{10}$, $\frac{1}{8784600}a^{18}-\frac{157}{8784600}a^{13}-\frac{24667}{798600}a^{8}+\frac{1}{10}a^{6}-\frac{1}{10}a^{5}+\frac{2191}{6600}a^{3}+\frac{1}{10}a-\frac{1}{10}$, $\frac{1}{96630600}a^{19}+\frac{259}{19326120}a^{14}-\frac{1}{36300}a^{13}+\frac{1}{3300}a^{12}-\frac{1}{3300}a^{11}+\frac{1}{3300}a^{10}+\frac{98357}{8784600}a^{9}+\frac{124}{3025}a^{8}+\frac{27}{550}a^{7}-\frac{137}{550}a^{6}+\frac{41}{275}a^{5}+\frac{30643}{72600}a^{4}-\frac{1601}{3300}a^{3}-\frac{49}{300}a^{2}-\frac{11}{300}a-\frac{19}{300}$ 1, a, a^2, a^3, a^4, 1/2*a^5 - 1/2, 1/2*a^6 - 1/2*a, 1/10*a^7 + 1/5*a^6 + 1/10*a^5 + 1/10*a^2 + 1/5*a + 1/10, 1/10*a^8 + 1/5*a^6 - 1/5*a^5 + 1/10*a^3 + 1/5*a - 1/5, 1/10*a^9 - 1/10*a^6 - 1/5*a^5 + 1/10*a^4 - 1/10*a - 1/5, 1/660*a^10 + 27/110*a^5 + 11/60, 1/660*a^11 + 27/110*a^6 + 11/60*a, 1/660*a^12 + 1/22*a^7 + 1/10*a^6 - 1/5*a^5 - 1/60*a^2 + 1/10*a - 1/5, 1/7260*a^13 - 3/605*a^8 - 1/10*a^6 + 1/10*a^5 - 247/660*a^3 - 1/10*a + 1/10, 1/36300*a^14 - 1/36300*a^13 + 1/3300*a^12 - 1/3300*a^11 + 1/3300*a^10 - 124/3025*a^9 + 124/3025*a^8 + 27/550*a^7 - 27/550*a^6 + 27/550*a^5 + 1601/3300*a^4 - 1601/3300*a^3 - 49/300*a^2 + 49/300*a - 49/300, 1/798600*a^15 - 157/798600*a^10 - 10147/72600*a^5 - 89/600, 1/798600*a^16 - 157/798600*a^11 - 10147/72600*a^6 - 89/600*a, 1/8784600*a^17 - 1367/8784600*a^12 - 20707/798600*a^7 + 1/5*a^6 + 1/10*a^5 - 1939/6600*a^2 + 1/5*a + 1/10, 1/8784600*a^18 - 157/8784600*a^13 - 24667/798600*a^8 + 1/10*a^6 - 1/10*a^5 + 2191/6600*a^3 + 1/10*a - 1/10, 1/96630600*a^19 + 259/19326120*a^14 - 1/36300*a^13 + 1/3300*a^12 - 1/3300*a^11 + 1/3300*a^10 + 98357/8784600*a^9 + 124/3025*a^8 + 27/550*a^7 - 137/550*a^6 + 41/275*a^5 + 30643/72600*a^4 - 1601/3300*a^3 - 49/300*a^2 - 11/300*a - 19/300

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:		$C_{5}$, which has order $5$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{5}$, which has order $5$ (assuming GRH)	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{7}{798600} a^{15} - \frac{1099}{798600} a^{10} + \frac{1571}{72600} a^{5} - \frac{623}{600} $ (7)/(798600)a^(15) - (1099)/(798600)a^(10) + (1571)/(72600)*a^(5) - (623)/(600) (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:	$\frac{1}{26620}a^{15}-\frac{9}{6655}a^{10}+\frac{21}{484}a^{5}-\frac{2}{5}$, $\frac{161}{96630600}a^{19}-\frac{1}{732050}a^{18}-\frac{8}{1098075}a^{17}-\frac{1}{199650}a^{16}-\frac{1}{72600}a^{15}+\frac{859}{96630600}a^{14}+\frac{193}{1464100}a^{13}+\frac{96}{366025}a^{12}-\frac{49}{399300}a^{11}+\frac{1}{24200}a^{10}-\frac{7}{1756920}a^{9}-\frac{212}{33275}a^{8}-\frac{457}{39930}a^{7}+\frac{107}{9075}a^{6}-\frac{41}{6600}a^{5}+\frac{8207}{72600}a^{4}+\frac{67}{1100}a^{3}-\frac{39}{550}a^{2}-\frac{19}{60}a-\frac{17}{40}$, $\frac{9}{8052550}a^{19}-\frac{3}{585640}a^{18}+\frac{8}{1098075}a^{17}-\frac{1}{72600}a^{16}+\frac{7}{798600}a^{15}-\frac{2303}{24157650}a^{14}+\frac{1741}{8784600}a^{13}-\frac{96}{366025}a^{12}+\frac{1}{24200}a^{11}-\frac{41}{53240}a^{10}+\frac{1743}{366025}a^{9}-\frac{2191}{266200}a^{8}+\frac{457}{39930}a^{7}-\frac{41}{6600}a^{6}+\frac{1439}{72600}a^{5}-\frac{761}{9075}a^{4}+\frac{581}{6600}a^{3}+\frac{39}{550}a^{2}-\frac{17}{40}a-\frac{93}{200}$, $\frac{71}{48315300}a^{19}+\frac{13}{2196150}a^{18}-\frac{3}{292820}a^{17}+\frac{1}{159720}a^{16}-\frac{1}{14520}a^{15}-\frac{213}{4026275}a^{14}-\frac{191}{1464100}a^{13}+\frac{2951}{4392300}a^{12}+\frac{61}{266200}a^{11}+\frac{49}{24200}a^{10}+\frac{281}{878460}a^{9}+\frac{1013}{199650}a^{8}-\frac{3511}{133100}a^{7}-\frac{179}{72600}a^{6}-\frac{241}{6600}a^{5}-\frac{63}{6050}a^{4}+\frac{29}{220}a^{3}+\frac{1411}{3300}a^{2}+\frac{21}{200}a-\frac{81}{200}$, $\frac{83}{48315300}a^{19}+\frac{1}{732050}a^{18}-\frac{17}{1756920}a^{17}-\frac{7}{798600}a^{16}+\frac{7}{798600}a^{15}-\frac{4319}{48315300}a^{14}-\frac{193}{1464100}a^{13}+\frac{1729}{8784600}a^{12}+\frac{41}{53240}a^{11}-\frac{41}{53240}a^{10}+\frac{8473}{4392300}a^{9}+\frac{212}{33275}a^{8}-\frac{5537}{798600}a^{7}-\frac{1439}{72600}a^{6}+\frac{1439}{72600}a^{5}+\frac{91}{7260}a^{4}-\frac{67}{1100}a^{3}-\frac{691}{6600}a^{2}+\frac{93}{200}a-\frac{93}{200}$, $\frac{629}{96630600}a^{19}-\frac{101}{8784600}a^{18}+\frac{29}{1756920}a^{17}+\frac{1}{798600}a^{16}-\frac{1}{19965}a^{15}-\frac{8003}{96630600}a^{14}+\frac{127}{8784600}a^{13}+\frac{287}{1756920}a^{12}-\frac{1367}{798600}a^{11}+\frac{53}{15972}a^{10}+\frac{79597}{8784600}a^{9}-\frac{12673}{798600}a^{8}+\frac{3529}{159720}a^{7}+\frac{1073}{72600}a^{6}-\frac{83}{1815}a^{5}-\frac{8971}{72600}a^{4}+\frac{899}{6600}a^{3}+\frac{331}{1320}a^{2}-\frac{1159}{600}a+\frac{269}{60}$, $\frac{169}{32210200}a^{19}+\frac{101}{8784600}a^{18}+\frac{1}{585640}a^{17}-\frac{17}{798600}a^{16}-\frac{19}{266200}a^{15}-\frac{4753}{32210200}a^{14}-\frac{127}{8784600}a^{13}+\frac{1223}{1756920}a^{12}+\frac{83}{266200}a^{11}-\frac{731}{798600}a^{10}+\frac{19197}{2928200}a^{9}+\frac{12673}{798600}a^{8}+\frac{21}{10648}a^{7}-\frac{1081}{72600}a^{6}-\frac{2347}{24200}a^{5}-\frac{6381}{24200}a^{4}-\frac{899}{6600}a^{3}+\frac{227}{264}a^{2}+\frac{251}{200}a-\frac{667}{600}$, $\frac{79}{96630600}a^{19}-\frac{1}{8784600}a^{18}+\frac{37}{2928200}a^{17}+\frac{13}{266200}a^{16}+\frac{1}{133100}a^{15}+\frac{3811}{96630600}a^{14}+\frac{1101}{2928200}a^{13}-\frac{1}{2928200}a^{12}-\frac{557}{798600}a^{11}+\frac{94}{99825}a^{10}+\frac{2143}{1756920}a^{9}-\frac{1601}{798600}a^{8}+\frac{5337}{266200}a^{7}+\frac{1893}{24200}a^{6}+\frac{61}{12100}a^{5}-\frac{457}{72600}a^{4}+\frac{19}{40}a^{3}+\frac{79}{440}a^{2}-\frac{137}{120}a-\frac{8}{15}$, $\frac{29}{8052550}a^{19}-\frac{1}{2196150}a^{18}-\frac{43}{1756920}a^{17}-\frac{1}{399300}a^{16}+\frac{13}{798600}a^{15}-\frac{757}{16105100}a^{14}+\frac{677}{4392300}a^{13}+\frac{6409}{8784600}a^{12}-\frac{17}{79860}a^{11}-\frac{519}{266200}a^{10}+\frac{2188}{366025}a^{9}+\frac{23}{39930}a^{8}-\frac{21607}{798600}a^{7}-\frac{677}{36300}a^{6}-\frac{1363}{72600}a^{5}+\frac{277}{12100}a^{4}+\frac{421}{3300}a^{3}+\frac{3329}{6600}a^{2}+\frac{127}{300}a-\frac{311}{200}$ 1/26620a^15 - 9/6655a^10 + 21/484a^5 - 2/5, 161/96630600a^19 - 1/732050a^18 - 8/1098075a^17 - 1/199650a^16 - 1/72600a^15 + 859/96630600a^14 + 193/1464100a^13 + 96/366025a^12 - 49/399300a^11 + 1/24200a^10 - 7/1756920a^9 - 212/33275a^8 - 457/39930a^7 + 107/9075a^6 - 41/6600a^5 + 8207/72600a^4 + 67/1100a^3 - 39/550a^2 - 19/60a - 17/40, 9/8052550a^19 - 3/585640a^18 + 8/1098075a^17 - 1/72600a^16 + 7/798600a^15 - 2303/24157650a^14 + 1741/8784600a^13 - 96/366025a^12 + 1/24200a^11 - 41/53240a^10 + 1743/366025a^9 - 2191/266200a^8 + 457/39930a^7 - 41/6600a^6 + 1439/72600a^5 - 761/9075a^4 + 581/6600a^3 + 39/550a^2 - 17/40a - 93/200, 71/48315300a^19 + 13/2196150a^18 - 3/292820a^17 + 1/159720a^16 - 1/14520a^15 - 213/4026275a^14 - 191/1464100a^13 + 2951/4392300a^12 + 61/266200a^11 + 49/24200a^10 + 281/878460a^9 + 1013/199650a^8 - 3511/133100a^7 - 179/72600a^6 - 241/6600a^5 - 63/6050a^4 + 29/220a^3 + 1411/3300a^2 + 21/200a - 81/200, 83/48315300a^19 + 1/732050a^18 - 17/1756920a^17 - 7/798600a^16 + 7/798600a^15 - 4319/48315300a^14 - 193/1464100a^13 + 1729/8784600a^12 + 41/53240a^11 - 41/53240a^10 + 8473/4392300a^9 + 212/33275a^8 - 5537/798600a^7 - 1439/72600a^6 + 1439/72600a^5 + 91/7260a^4 - 67/1100a^3 - 691/6600a^2 + 93/200a - 93/200, 629/96630600a^19 - 101/8784600a^18 + 29/1756920a^17 + 1/798600a^16 - 1/19965a^15 - 8003/96630600a^14 + 127/8784600a^13 + 287/1756920a^12 - 1367/798600a^11 + 53/15972a^10 + 79597/8784600a^9 - 12673/798600a^8 + 3529/159720a^7 + 1073/72600a^6 - 83/1815a^5 - 8971/72600a^4 + 899/6600a^3 + 331/1320a^2 - 1159/600a + 269/60, 169/32210200a^19 + 101/8784600a^18 + 1/585640a^17 - 17/798600a^16 - 19/266200a^15 - 4753/32210200a^14 - 127/8784600a^13 + 1223/1756920a^12 + 83/266200a^11 - 731/798600a^10 + 19197/2928200a^9 + 12673/798600a^8 + 21/10648a^7 - 1081/72600a^6 - 2347/24200a^5 - 6381/24200a^4 - 899/6600a^3 + 227/264a^2 + 251/200a - 667/600, 79/96630600a^19 - 1/8784600a^18 + 37/2928200a^17 + 13/266200a^16 + 1/133100a^15 + 3811/96630600a^14 + 1101/2928200a^13 - 1/2928200a^12 - 557/798600a^11 + 94/99825a^10 + 2143/1756920a^9 - 1601/798600a^8 + 5337/266200a^7 + 1893/24200a^6 + 61/12100a^5 - 457/72600a^4 + 19/40a^3 + 79/440a^2 - 137/120a - 8/15, 29/8052550a^19 - 1/2196150a^18 - 43/1756920a^17 - 1/399300a^16 + 13/798600a^15 - 757/16105100a^14 + 677/4392300a^13 + 6409/8784600a^12 - 17/79860a^11 - 519/266200a^10 + 2188/366025a^9 + 23/39930a^8 - 21607/798600a^7 - 677/36300a^6 - 1363/72600a^5 + 277/12100a^4 + 421/3300a^3 + 3329/6600a^2 + 127/300*a - 311/200 (assuming GRH)	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:	$ 45537996.331 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 45537996.331 \cdot 5}{10\cdot\sqrt{23383113568432629108428955078125}}\cr\approx \mathstrut & 0.45153501728 \end{aligned}\] (assuming GRH)

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 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561) 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 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561, 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 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561); 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 - 36*x^15 + 2486*x^10 - 47916*x^5 + 1771561); 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

$C_5:F_5$ (as 20T26):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

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

A solvable group of order 100

The 13 conjugacy class representatives for $C_5:F_5$

Character table for $C_5:F_5$

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 25 sibling:	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	${\href{/padicField/13.4.0.1}{4} }^{5}$	${\href{/padicField/17.4.0.1}{4} }^{5}$	${\href{/padicField/19.10.0.1}{10} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{5}$	${\href{/padicField/29.10.0.1}{10} }^{2}$	${\href{/padicField/31.5.0.1}{5} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{5}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.5.0.1}{5} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{5}$	${\href{/padicField/43.4.0.1}{4} }^{5}$	${\href{/padicField/47.4.0.1}{4} }^{5}$	${\href{/padicField/53.4.0.1}{4} }^{5}$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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	11.1.5.4a1.4	$x^{5} + 55$	$5$	$1$	$4$	$C_5$	$$[\ ]_{5}$$
	11.1.5.4a1.3	$x^{5} + 44$	$5$	$1$	$4$	$C_5$	$$[\ ]_{5}$$
	11.1.5.4a1.2	$x^{5} + 22$	$5$	$1$	$4$	$C_5$	$$[\ ]_{5}$$
	11.5.1.0a1.1	$x^{5} + 10 x^{2} + 9$	$1$	$5$	$0$	$C_5$	$$[\ ]^{5}$$

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