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Properties

Label 20.10.185...896.1
Degree $20$
Signature $[10, 5]$
Discriminant $-1.855\times 10^{25}$
Root discriminant \(18.34\)
Ramified primes $2,3,17,4153$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.C_2^5.S_5$ (as 20T992)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9)
 
gp: K = bnfinit(y^20 - 7*y^18 + 18*y^16 - 38*y^14 + 84*y^12 - 60*y^10 - 128*y^8 + 261*y^6 - 186*y^4 + 63*y^2 - 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9)
 

\( x^{20} - 7x^{18} + 18x^{16} - 38x^{14} + 84x^{12} - 60x^{10} - 128x^{8} + 261x^{6} - 186x^{4} + 63x^{2} - 9 \) 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:  $[10, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-18546881403289244170576896\) \(\medspace = -\,2^{10}\cdot 3^{6}\cdot 17^{4}\cdot 4153^{4}\) 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:  \(18.34\)
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:  $2^{31/16}3^{3/4}17^{1/2}4153^{1/2}\approx 2320.0213627026787$
Ramified primes:   \(2\), \(3\), \(17\), \(4153\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{6}a^{16}-\frac{1}{6}a^{14}-\frac{1}{2}a^{13}+\frac{1}{6}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{3}a^{4}-\frac{1}{2}a$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{15}-\frac{1}{2}a^{14}+\frac{1}{6}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4114794}a^{18}+\frac{292721}{4114794}a^{16}+\frac{193775}{685799}a^{14}-\frac{1}{2}a^{13}+\frac{741314}{2057397}a^{12}-\frac{1}{2}a^{11}+\frac{296505}{1371598}a^{10}+\frac{297090}{685799}a^{8}-\frac{1}{2}a^{7}+\frac{284455}{4114794}a^{6}-\frac{1}{2}a^{5}-\frac{180880}{685799}a^{4}-\frac{1}{2}a^{3}-\frac{157278}{685799}a^{2}+\frac{139787}{1371598}$, $\frac{1}{4114794}a^{19}+\frac{292721}{4114794}a^{17}-\frac{298249}{1371598}a^{15}-\frac{1}{2}a^{14}-\frac{574769}{4114794}a^{13}+\frac{296505}{1371598}a^{11}-\frac{91619}{1371598}a^{9}-\frac{886471}{2057397}a^{7}+\frac{324039}{1371598}a^{5}-\frac{1}{2}a^{4}-\frac{157278}{685799}a^{3}+\frac{139787}{1371598}a-\frac{1}{2}$ 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 $1$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{150702}{685799}a^{18}-\frac{2513197}{2057397}a^{16}+\frac{4225159}{2057397}a^{14}-\frac{3109937}{685799}a^{12}+\frac{21051563}{2057397}a^{10}+\frac{3503182}{685799}a^{8}-\frac{19874653}{685799}a^{6}+\frac{32385413}{2057397}a^{4}+\frac{797895}{685799}a^{2}-\frac{579624}{685799}$, $\frac{237524}{2057397}a^{18}-\frac{1468811}{2057397}a^{16}+\frac{1055425}{685799}a^{14}-\frac{6968815}{2057397}a^{12}+\frac{5097506}{685799}a^{10}-\frac{1309086}{685799}a^{8}-\frac{28831618}{2057397}a^{6}+\frac{12847847}{685799}a^{4}-\frac{8456877}{685799}a^{2}+\frac{1866200}{685799}$, $\frac{1391671}{2057397}a^{18}-\frac{9052586}{2057397}a^{16}+\frac{20707450}{2057397}a^{14}-\frac{43346897}{2057397}a^{12}+\frac{96469490}{2057397}a^{10}-\frac{12785450}{685799}a^{8}-\frac{191435180}{2057397}a^{6}+\frac{273213596}{2057397}a^{4}-\frac{45879728}{685799}a^{2}+\frac{7884531}{685799}$, $\frac{51981}{685799}a^{18}-\frac{1047934}{2057397}a^{16}+\frac{2447320}{2057397}a^{14}-\frac{1605552}{685799}a^{12}+\frac{10789895}{2057397}a^{10}-\frac{1462748}{685799}a^{8}-\frac{8486472}{685799}a^{6}+\frac{35319728}{2057397}a^{4}-\frac{3090230}{685799}a^{2}-\frac{1374471}{685799}$, $\frac{122417}{2057397}a^{19}+\frac{831965}{2057397}a^{18}-\frac{342691}{2057397}a^{17}-\frac{3275879}{1371598}a^{16}-\frac{1769825}{4114794}a^{15}+\frac{19075169}{4114794}a^{14}+\frac{1480727}{2057397}a^{13}-\frac{41706851}{4114794}a^{12}-\frac{2836460}{2057397}a^{11}+\frac{93246613}{4114794}a^{10}+\frac{7377512}{685799}a^{9}+\frac{700118}{685799}a^{8}-\frac{15775666}{2057397}a^{7}-\frac{106585288}{2057397}a^{6}-\frac{74001463}{4114794}a^{5}+\frac{97827869}{2057397}a^{4}+\frac{14327978}{685799}a^{3}-\frac{13085919}{685799}a^{2}-\frac{9789923}{1371598}a+\frac{4994663}{1371598}$, $\frac{421369}{4114794}a^{19}+\frac{12695}{1371598}a^{18}-\frac{2881493}{4114794}a^{17}-\frac{69259}{4114794}a^{16}+\frac{3447698}{2057397}a^{15}-\frac{458783}{4114794}a^{14}-\frac{13637891}{4114794}a^{13}+\frac{208905}{1371598}a^{12}+\frac{15532363}{2057397}a^{11}-\frac{1291825}{4114794}a^{10}-\frac{2604848}{685799}a^{9}+\frac{2778893}{1371598}a^{8}-\frac{33604816}{2057397}a^{7}-\frac{473554}{685799}a^{6}+\frac{93723725}{4114794}a^{5}-\frac{15556447}{4114794}a^{4}-\frac{11433017}{1371598}a^{3}+\frac{1207434}{685799}a^{2}+\frac{1375489}{1371598}a-\frac{755541}{1371598}$, $\frac{421369}{4114794}a^{19}-\frac{12695}{1371598}a^{18}-\frac{2881493}{4114794}a^{17}+\frac{69259}{4114794}a^{16}+\frac{3447698}{2057397}a^{15}+\frac{458783}{4114794}a^{14}-\frac{13637891}{4114794}a^{13}-\frac{208905}{1371598}a^{12}+\frac{15532363}{2057397}a^{11}+\frac{1291825}{4114794}a^{10}-\frac{2604848}{685799}a^{9}-\frac{2778893}{1371598}a^{8}-\frac{33604816}{2057397}a^{7}+\frac{473554}{685799}a^{6}+\frac{93723725}{4114794}a^{5}+\frac{15556447}{4114794}a^{4}-\frac{11433017}{1371598}a^{3}-\frac{1207434}{685799}a^{2}+\frac{1375489}{1371598}a+\frac{755541}{1371598}$, $\frac{3905687}{4114794}a^{19}-\frac{421369}{4114794}a^{18}-\frac{12540382}{2057397}a^{17}+\frac{2881493}{4114794}a^{16}+\frac{18594955}{1371598}a^{15}-\frac{3447698}{2057397}a^{14}-\frac{116010895}{4114794}a^{13}+\frac{13637891}{4114794}a^{12}+\frac{86855631}{1371598}a^{11}-\frac{15532363}{2057397}a^{10}-\frac{27738591}{1371598}a^{9}+\frac{2604848}{685799}a^{8}-\frac{549452017}{4114794}a^{7}+\frac{33604816}{2057397}a^{6}+\frac{234032883}{1371598}a^{5}-\frac{93723725}{4114794}a^{4}-\frac{104047845}{1371598}a^{3}+\frac{11433017}{1371598}a^{2}+\frac{18887141}{1371598}a-\frac{1375489}{1371598}$, $\frac{450116}{685799}a^{19}-\frac{1885375}{4114794}a^{18}-\frac{2784636}{685799}a^{17}+\frac{5659435}{2057397}a^{16}+\frac{5809083}{685799}a^{15}-\frac{22187035}{4114794}a^{14}-\frac{12195429}{685799}a^{13}+\frac{23367421}{2057397}a^{12}+\frac{54958045}{1371598}a^{11}-\frac{53154097}{2057397}a^{10}-\frac{7703011}{1371598}a^{9}-\frac{1508397}{1371598}a^{8}-\frac{62648430}{685799}a^{7}+\frac{131367289}{2057397}a^{6}+\frac{134288823}{1371598}a^{5}-\frac{114387226}{2057397}a^{4}-\frac{49027435}{1371598}a^{3}+\frac{9781419}{685799}a^{2}+\frac{6987227}{1371598}a-\frac{1102621}{1371598}$, $\frac{939275}{2057397}a^{19}-\frac{3640261}{4114794}a^{18}-\frac{12093007}{4114794}a^{17}+\frac{23247205}{4114794}a^{16}+\frac{9093869}{1371598}a^{15}-\frac{17112263}{1371598}a^{14}-\frac{57679481}{4114794}a^{13}+\frac{53594830}{2057397}a^{12}+\frac{21448739}{685799}a^{11}-\frac{40048677}{685799}a^{10}-\frac{15785591}{1371598}a^{9}+\frac{23991643}{1371598}a^{8}-\frac{125835100}{2057397}a^{7}+\frac{507701801}{4114794}a^{6}+\frac{115941493}{1371598}a^{5}-\frac{107282442}{685799}a^{4}-\frac{63845943}{1371598}a^{3}+\frac{96390857}{1371598}a^{2}+\frac{8388066}{685799}a-\frac{16137181}{1371598}$, $\frac{1061692}{2057397}a^{19}+\frac{554815}{1371598}a^{18}-\frac{4259463}{1371598}a^{17}-\frac{5661850}{2057397}a^{16}+\frac{12755891}{2057397}a^{15}+\frac{13683490}{2057397}a^{14}-\frac{54718027}{4114794}a^{13}-\frac{18616499}{1371598}a^{12}+\frac{61509757}{2057397}a^{11}+\frac{125465659}{4114794}a^{10}-\frac{1030567}{1371598}a^{9}-\frac{22466383}{1371598}a^{8}-\frac{141610766}{2057397}a^{7}-\frac{81204131}{1371598}a^{6}+\frac{136911508}{2057397}a^{5}+\frac{188699729}{2057397}a^{4}-\frac{35189987}{1371598}a^{3}-\frac{64038559}{1371598}a^{2}+\frac{6986209}{1371598}a+\frac{6259931}{685799}$, $\frac{3640261}{4114794}a^{19}+\frac{939565}{4114794}a^{18}-\frac{23247205}{4114794}a^{17}-\frac{6539389}{4114794}a^{16}+\frac{17112263}{1371598}a^{15}+\frac{5494097}{1371598}a^{14}-\frac{53594830}{2057397}a^{13}-\frac{17008543}{2057397}a^{12}+\frac{40048677}{685799}a^{11}+\frac{25139309}{1371598}a^{10}-\frac{23991643}{1371598}a^{9}-\frac{8144316}{685799}a^{8}-\frac{507701801}{4114794}a^{7}-\frac{131811221}{4114794}a^{6}+\frac{107282442}{685799}a^{5}+\frac{80276061}{1371598}a^{4}-\frac{96390857}{1371598}a^{3}-\frac{23681711}{685799}a^{2}+\frac{16137181}{1371598}a+\frac{4574977}{685799}$, $\frac{1742159}{2057397}a^{19}+\frac{229727}{2057397}a^{18}-\frac{7399757}{1371598}a^{17}-\frac{491792}{685799}a^{16}+\frac{48889469}{4114794}a^{15}+\frac{6436613}{4114794}a^{14}-\frac{51186502}{2057397}a^{13}-\frac{6505588}{2057397}a^{12}+\frac{229502167}{4114794}a^{11}+\frac{29772901}{4114794}a^{10}-\frac{22528895}{1371598}a^{9}-\frac{2430803}{1371598}a^{8}-\frac{482242385}{4114794}a^{7}-\frac{35025478}{2057397}a^{6}+\frac{304187462}{2057397}a^{5}+\frac{39083639}{2057397}a^{4}-\frac{46307414}{685799}a^{3}-\frac{9018149}{1371598}a^{2}+\frac{8070027}{685799}a+\frac{995773}{685799}$, $\frac{288656}{2057397}a^{19}-\frac{2226497}{2057397}a^{18}-\frac{1309809}{1371598}a^{17}+\frac{14195102}{2057397}a^{16}+\frac{4671358}{2057397}a^{15}-\frac{20764889}{1371598}a^{14}-\frac{18454547}{4114794}a^{13}+\frac{129508457}{4114794}a^{12}+\frac{41854621}{4114794}a^{11}-\frac{97158879}{1371598}a^{10}-\frac{3336222}{685799}a^{9}+\frac{13603671}{685799}a^{8}-\frac{46334524}{2057397}a^{7}+\frac{628708805}{4114794}a^{6}+\frac{129043453}{4114794}a^{5}-\frac{128585012}{685799}a^{4}-\frac{6918724}{685799}a^{3}+\frac{109368763}{1371598}a^{2}+\frac{509}{685799}a-\frac{18240109}{1371598}$ Copy content Toggle raw display (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:  \( 136001.268357 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 136001.268357 \cdot 1}{2\cdot\sqrt{18546881403289244170576896}}\cr\approx \mathstrut & 0.158334959006 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9)
 
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 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9, 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 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9);
 
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 - 7*x^18 + 18*x^16 - 38*x^14 + 84*x^12 - 60*x^10 - 128*x^8 + 261*x^6 - 186*x^4 + 63*x^2 - 9);
 
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

$C_2^9.C_2^5.S_5$ (as 20T992):

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 1966080
The 280 conjugacy class representatives for $C_2^9.C_2^5.S_5$
Character table for $C_2^9.C_2^5.S_5$

Intermediate fields

5.5.70601.1, 10.6.44860510809.1

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 siblings: data not computed
Degree 40 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 R R ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ R ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ $16{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.10$x^{10} + 10 x^{9} + 22 x^{8} + 160 x^{6} + 1200 x^{5} + 3168 x^{4} + 4480 x^{3} + 3760 x^{2} + 1824 x + 416$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
\(3\) Copy content Toggle raw display 3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.6.3$x^{8} - 6 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.6.0.1$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
\(4153\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$