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Properties

Label 20.0.250...000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.500\times 10^{21}$
Root discriminant \(11.75\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $C_5\times D_{10}$ (as 20T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1)
 
gp: K = bnfinit(y^20 - 4*y^19 + 7*y^18 - 6*y^17 - 2*y^16 + 6*y^15 + 14*y^14 - 54*y^13 + 64*y^12 + 24*y^11 - 190*y^10 + 290*y^9 - 169*y^8 - 154*y^7 + 464*y^6 - 554*y^5 + 417*y^4 - 208*y^3 + 66*y^2 - 12*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1)
 

\( x^{20} - 4 x^{19} + 7 x^{18} - 6 x^{17} - 2 x^{16} + 6 x^{15} + 14 x^{14} - 54 x^{13} + 64 x^{12} + \cdots + 1 \) Copy content Toggle raw display

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:   \(2500000000000000000000\) \(\medspace = 2^{20}\cdot 5^{22}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(11.75\)
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:  $2\cdot 5^{17/10}\approx 30.85169313600048$
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}-\frac{1}{7}a^{15}-\frac{2}{7}a^{14}-\frac{3}{7}a^{13}+\frac{2}{7}a^{11}-\frac{2}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}-\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{17}-\frac{3}{7}a^{15}+\frac{2}{7}a^{14}-\frac{3}{7}a^{13}+\frac{2}{7}a^{12}+\frac{2}{7}a^{11}-\frac{2}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{18}-\frac{1}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{12}-\frac{3}{7}a^{11}-\frac{1}{7}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{3}{7}a^{6}-\frac{1}{7}a^{5}-\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{22230187}a^{19}+\frac{316742}{22230187}a^{18}-\frac{1248133}{22230187}a^{17}+\frac{72912}{3175741}a^{16}-\frac{5356625}{22230187}a^{15}-\frac{2276879}{22230187}a^{14}+\frac{369525}{3175741}a^{13}+\frac{9538906}{22230187}a^{12}+\frac{10332540}{22230187}a^{11}+\frac{1228161}{3175741}a^{10}-\frac{222459}{22230187}a^{9}-\frac{6008298}{22230187}a^{8}+\frac{9247629}{22230187}a^{7}-\frac{1567752}{22230187}a^{6}-\frac{4433063}{22230187}a^{5}+\frac{8436821}{22230187}a^{4}+\frac{9943944}{22230187}a^{3}+\frac{1386475}{22230187}a^{2}-\frac{109510}{22230187}a-\frac{201610}{3175741}$ Copy content Toggle raw display

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 $1$

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:   \( -\frac{57119}{9107} a^{19} + \frac{26866}{1301} a^{18} - \frac{267410}{9107} a^{17} + \frac{156983}{9107} a^{16} + \frac{218307}{9107} a^{15} - \frac{181352}{9107} a^{14} - \frac{926865}{9107} a^{13} + \frac{2419212}{9107} a^{12} - \frac{1951750}{9107} a^{11} - \frac{2701613}{9107} a^{10} + \frac{1266459}{1301} a^{9} - \frac{10294686}{9107} a^{8} + \frac{2557279}{9107} a^{7} + \frac{10292841}{9107} a^{6} - \frac{19014662}{9107} a^{5} + \frac{18350921}{9107} a^{4} - \frac{11338829}{9107} a^{3} + \frac{4435672}{9107} a^{2} - \frac{1011806}{9107} a + \frac{107027}{9107} \)  (order $4$) Copy content Toggle raw display
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:   $\frac{18912006}{3175741}a^{19}-\frac{469207801}{22230187}a^{18}+\frac{696786026}{22230187}a^{17}-\frac{428962951}{22230187}a^{16}-\frac{520610511}{22230187}a^{15}+\frac{579232476}{22230187}a^{14}+\frac{2193417843}{22230187}a^{13}-\frac{882894234}{3175741}a^{12}+\frac{5381288298}{22230187}a^{11}+\frac{6240563925}{22230187}a^{10}-\frac{22659122460}{22230187}a^{9}+\frac{27131369652}{22230187}a^{8}-\frac{7737767500}{22230187}a^{7}-\frac{26068255952}{22230187}a^{6}+\frac{49358820704}{22230187}a^{5}-\frac{47734643902}{22230187}a^{4}+\frac{4147553537}{3175741}a^{3}-\frac{10830306582}{22230187}a^{2}+\frac{2158231877}{22230187}a-\frac{186884032}{22230187}$, $\frac{699802}{3175741}a^{19}-\frac{22620897}{22230187}a^{18}+\frac{52966113}{22230187}a^{17}-\frac{64451337}{22230187}a^{16}+\frac{17210346}{22230187}a^{15}+\frac{48013919}{22230187}a^{14}+\frac{14226421}{22230187}a^{13}-\frac{326622522}{22230187}a^{12}+\frac{596942336}{22230187}a^{11}-\frac{231490934}{22230187}a^{10}-\frac{154864866}{3175741}a^{9}+\frac{2486417714}{22230187}a^{8}-\frac{2263334792}{22230187}a^{7}-\frac{168318967}{22230187}a^{6}+\frac{3514112691}{22230187}a^{5}-\frac{5152925334}{22230187}a^{4}+\frac{4346126546}{22230187}a^{3}-\frac{2237892887}{22230187}a^{2}+\frac{672207672}{22230187}a-\frac{82141887}{22230187}$, $\frac{52251229}{22230187}a^{19}-\frac{20797040}{3175741}a^{18}+\frac{23654941}{3175741}a^{17}-\frac{46149449}{22230187}a^{16}-\frac{238074483}{22230187}a^{15}+\frac{45270500}{22230187}a^{14}+\frac{128707562}{3175741}a^{13}-\frac{1755538609}{22230187}a^{12}+\frac{803192232}{22230187}a^{11}+\frac{3049490289}{22230187}a^{10}-\frac{6612933162}{22230187}a^{9}+\frac{5702780623}{22230187}a^{8}+\frac{1190275056}{22230187}a^{7}-\frac{9241374918}{22230187}a^{6}+\frac{1768124500}{3175741}a^{5}-\frac{1358243418}{3175741}a^{4}+\frac{4498899555}{22230187}a^{3}-\frac{1184570064}{22230187}a^{2}+\frac{189536024}{22230187}a-\frac{32014916}{22230187}$, $\frac{8173423}{3175741}a^{19}-\frac{158673093}{22230187}a^{18}+\frac{178147573}{22230187}a^{17}-\frac{41944177}{22230187}a^{16}-\frac{271368593}{22230187}a^{15}+\frac{58237316}{22230187}a^{14}+\frac{141082189}{3175741}a^{13}-\frac{1930109845}{22230187}a^{12}+\frac{835795157}{22230187}a^{11}+\frac{3425910797}{22230187}a^{10}-\frac{7323623117}{22230187}a^{9}+\frac{6156286420}{22230187}a^{8}+\frac{1712114974}{22230187}a^{7}-\frac{10622219810}{22230187}a^{6}+\frac{13771978233}{22230187}a^{5}-\frac{9903826563}{22230187}a^{4}+\frac{3902205409}{22230187}a^{3}-\frac{275817383}{22230187}a^{2}-\frac{49290959}{3175741}a+\frac{78141851}{22230187}$, $\frac{9873327}{22230187}a^{19}-\frac{29487821}{22230187}a^{18}+\frac{40806067}{22230187}a^{17}-\frac{26879376}{22230187}a^{16}-\frac{31136725}{22230187}a^{15}+\frac{16848252}{22230187}a^{14}+\frac{147381681}{22230187}a^{13}-\frac{361219735}{22230187}a^{12}+\frac{296694275}{22230187}a^{11}+\frac{405535329}{22230187}a^{10}-\frac{1319217447}{22230187}a^{9}+\frac{1605215200}{22230187}a^{8}-\frac{521419289}{22230187}a^{7}-\frac{1408159952}{22230187}a^{6}+\frac{2897540052}{22230187}a^{5}-\frac{3086956584}{22230187}a^{4}+\frac{302056462}{3175741}a^{3}-\frac{140107556}{3175741}a^{2}+\frac{248327593}{22230187}a-\frac{646188}{3175741}$, $\frac{28897571}{22230187}a^{19}-\frac{93912633}{22230187}a^{18}+\frac{120266788}{22230187}a^{17}-\frac{51856058}{22230187}a^{16}-\frac{127394468}{22230187}a^{15}+\frac{11763251}{3175741}a^{14}+\frac{513062146}{22230187}a^{13}-\frac{1174552146}{22230187}a^{12}+\frac{764344487}{22230187}a^{11}+\frac{1613647156}{22230187}a^{10}-\frac{4381793009}{22230187}a^{9}+\frac{4429224405}{22230187}a^{8}-\frac{264181599}{22230187}a^{7}-\frac{5671056780}{22230187}a^{6}+\frac{8837479629}{22230187}a^{5}-\frac{7593446029}{22230187}a^{4}+\frac{4000439329}{22230187}a^{3}-\frac{1125834102}{22230187}a^{2}+\frac{65609254}{22230187}a+\frac{22439191}{22230187}$, $\frac{2268820}{22230187}a^{19}+\frac{27086701}{22230187}a^{18}-\frac{85662813}{22230187}a^{17}+\frac{106961244}{22230187}a^{16}-\frac{47741643}{22230187}a^{15}-\frac{146071993}{22230187}a^{14}+\frac{73513845}{22230187}a^{13}+\frac{481426565}{22230187}a^{12}-\frac{156348383}{3175741}a^{11}+\frac{713690426}{22230187}a^{10}+\frac{1585783509}{22230187}a^{9}-\frac{4013240642}{22230187}a^{8}+\frac{3940548769}{22230187}a^{7}+\frac{76547561}{22230187}a^{6}-\frac{5344137224}{22230187}a^{5}+\frac{7842683817}{22230187}a^{4}-\frac{6355450336}{22230187}a^{3}+\frac{3101351468}{22230187}a^{2}-\frac{791964833}{22230187}a+\frac{83600929}{22230187}$, $\frac{8133218}{3175741}a^{19}-\frac{199192626}{22230187}a^{18}+\frac{292073094}{22230187}a^{17}-\frac{176480909}{22230187}a^{16}-\frac{218931267}{22230187}a^{15}+\frac{228501197}{22230187}a^{14}+\frac{946872977}{22230187}a^{13}-\frac{2601304366}{22230187}a^{12}+\frac{2203257580}{22230187}a^{11}+\frac{382961071}{3175741}a^{10}-\frac{9497815111}{22230187}a^{9}+\frac{11275402905}{22230187}a^{8}-\frac{3160444828}{22230187}a^{7}-\frac{10788650057}{22230187}a^{6}+\frac{20521079744}{22230187}a^{5}-\frac{19997248903}{22230187}a^{4}+\frac{12492825309}{22230187}a^{3}-\frac{4946627349}{22230187}a^{2}+\frac{1140590520}{22230187}a-\frac{19179999}{3175741}$, $\frac{86489267}{22230187}a^{19}-\frac{279494459}{22230187}a^{18}+\frac{54063787}{3175741}a^{17}-\frac{195232594}{22230187}a^{16}-\frac{353777804}{22230187}a^{15}+\frac{242668081}{22230187}a^{14}+\frac{1459976232}{22230187}a^{13}-\frac{3547291975}{22230187}a^{12}+\frac{368115137}{3175741}a^{11}+\frac{4435405692}{22230187}a^{10}-\frac{1868254249}{3175741}a^{9}+\frac{14198364703}{22230187}a^{8}-\frac{2274031971}{22230187}a^{7}-\frac{15932916602}{22230187}a^{6}+\frac{27078442070}{22230187}a^{5}-\frac{24882965151}{22230187}a^{4}+\frac{14658774374}{22230187}a^{3}-\frac{5426156740}{22230187}a^{2}+\frac{1208261745}{22230187}a-\frac{150140205}{22230187}$ Copy content Toggle raw display
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:  \( 397.737246993 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 397.737246993 \cdot 1}{4\cdot\sqrt{2500000000000000000000}}\cr\approx \mathstrut & 0.190706260938 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*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^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*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^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*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^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*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

$C_5\times D_{10}$ (as 20T24):

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 100
The 40 conjugacy class representatives for $C_5\times D_{10}$
Character table for $C_5\times D_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 10.0.400000000.1

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 20 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 R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.2.0.1}{2} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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

# 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $20$$2$$10$$20$
\(5\) Copy content Toggle raw display 5.10.17.5$x^{10} + 20 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$