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Properties

Label 20.4.109...641.2
Degree $20$
Signature $[4, 8]$
Discriminant $1.091\times 10^{23}$
Root discriminant \(14.19\)
Ramified primes $11,23,67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}:C_5$ (as 20T341)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^20 - y^19 + 6*y^18 - 8*y^17 + 18*y^16 - 13*y^15 + 25*y^14 + 10*y^13 - 23*y^12 + 65*y^11 - 103*y^10 + 107*y^9 - 130*y^8 + 94*y^7 - 65*y^6 + 53*y^5 - 12*y^4 + 5*y^3 - 3*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*x + 1)
 

\( x^{20} - x^{19} + 6 x^{18} - 8 x^{17} + 18 x^{16} - 13 x^{15} + 25 x^{14} + 10 x^{13} - 23 x^{12} + \cdots + 1 \) 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:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(109115950462157402855641\) \(\medspace = 11^{16}\cdot 23^{2}\cdot 67^{2}\) 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:  \(14.19\)
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:  $11^{4/5}23^{1/2}67^{1/2}\approx 267.310161002235$
Ramified primes:   \(11\), \(23\), \(67\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{24043381161601}a^{19}-\frac{887723158630}{24043381161601}a^{18}-\frac{11325102968358}{24043381161601}a^{17}-\frac{1760672281189}{24043381161601}a^{16}-\frac{10203571034040}{24043381161601}a^{15}-\frac{2520353473565}{24043381161601}a^{14}+\frac{8775403047978}{24043381161601}a^{13}+\frac{2309411901077}{24043381161601}a^{12}+\frac{6443082535796}{24043381161601}a^{11}+\frac{1619142654070}{24043381161601}a^{10}+\frac{7173455697132}{24043381161601}a^{9}+\frac{4174553474737}{24043381161601}a^{8}+\frac{4545464710198}{24043381161601}a^{7}-\frac{11100365907521}{24043381161601}a^{6}-\frac{5029093494082}{24043381161601}a^{5}+\frac{4986812363911}{24043381161601}a^{4}-\frac{2735597482262}{24043381161601}a^{3}-\frac{6771352547386}{24043381161601}a^{2}+\frac{3742661249908}{24043381161601}a-\frac{8402550420312}{24043381161601}$ 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:  $11$
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:   $\frac{8310573784258}{24043381161601}a^{19}-\frac{6022105564950}{24043381161601}a^{18}+\frac{40980594393456}{24043381161601}a^{17}-\frac{43886059021100}{24043381161601}a^{16}+\frac{92260407181975}{24043381161601}a^{15}-\frac{7059277520374}{24043381161601}a^{14}+\frac{61153158572119}{24043381161601}a^{13}+\frac{224706441839121}{24043381161601}a^{12}-\frac{290478096156996}{24043381161601}a^{11}+\frac{393392995429716}{24043381161601}a^{10}-\frac{453433903242612}{24043381161601}a^{9}+\frac{160169690903265}{24043381161601}a^{8}-\frac{171785712448909}{24043381161601}a^{7}-\frac{158375871908996}{24043381161601}a^{6}+\frac{274975826286504}{24043381161601}a^{5}-\frac{110617711583550}{24043381161601}a^{4}+\frac{158984457823382}{24043381161601}a^{3}-\frac{90695952960481}{24043381161601}a^{2}+\frac{540245568405}{24043381161601}a-\frac{10121237633514}{24043381161601}$, $\frac{7784016854642}{24043381161601}a^{19}-\frac{9484420171746}{24043381161601}a^{18}+\frac{47836161120880}{24043381161601}a^{17}-\frac{71841110989010}{24043381161601}a^{16}+\frac{150991301003502}{24043381161601}a^{15}-\frac{127255254705036}{24043381161601}a^{14}+\frac{207591315579099}{24043381161601}a^{13}+\frac{41923325219561}{24043381161601}a^{12}-\frac{212272606651527}{24043381161601}a^{11}+\frac{549828858370590}{24043381161601}a^{10}-\frac{910370356166046}{24043381161601}a^{9}+\frac{981429161196328}{24043381161601}a^{8}-\frac{11\!\cdots\!71}{24043381161601}a^{7}+\frac{856809690660639}{24043381161601}a^{6}-\frac{566158421380867}{24043381161601}a^{5}+\frac{400995895252090}{24043381161601}a^{4}-\frac{131799255490737}{24043381161601}a^{3}+\frac{19177607440280}{24043381161601}a^{2}-\frac{5850107759051}{24043381161601}a-\frac{17444697254273}{24043381161601}$, $\frac{7882374331898}{24043381161601}a^{19}-\frac{5765582143889}{24043381161601}a^{18}+\frac{40536143851283}{24043381161601}a^{17}-\frac{47681596431846}{24043381161601}a^{16}+\frac{101555148940523}{24043381161601}a^{15}-\frac{43032609808997}{24043381161601}a^{14}+\frac{114448462744525}{24043381161601}a^{13}+\frac{131967529805524}{24043381161601}a^{12}-\frac{215354410718215}{24043381161601}a^{11}+\frac{337727526970142}{24043381161601}a^{10}-\frac{567860131500347}{24043381161601}a^{9}+\frac{392372520887759}{24043381161601}a^{8}-\frac{588531126770809}{24043381161601}a^{7}+\frac{304046895298718}{24043381161601}a^{6}-\frac{164142857656398}{24043381161601}a^{5}+\frac{262305463984334}{24043381161601}a^{4}-\frac{32794958802655}{24043381161601}a^{3}+\frac{5700852281599}{24043381161601}a^{2}-\frac{43406081135893}{24043381161601}a-\frac{16715037373353}{24043381161601}$, $\frac{5372896308853}{24043381161601}a^{19}-\frac{7932672843955}{24043381161601}a^{18}+\frac{40173969027825}{24043381161601}a^{17}-\frac{62584581341212}{24043381161601}a^{16}+\frac{146073587237824}{24043381161601}a^{15}-\frac{148603572731299}{24043381161601}a^{14}+\frac{242697443262068}{24043381161601}a^{13}-\frac{37455594073588}{24043381161601}a^{12}-\frac{65998606899994}{24043381161601}a^{11}+\frac{520734438095688}{24043381161601}a^{10}-\frac{859193542722143}{24043381161601}a^{9}+\frac{11\!\cdots\!02}{24043381161601}a^{8}-\frac{13\!\cdots\!46}{24043381161601}a^{7}+\frac{11\!\cdots\!66}{24043381161601}a^{6}-\frac{929628942913319}{24043381161601}a^{5}+\frac{639437858303498}{24043381161601}a^{4}-\frac{256306987061412}{24043381161601}a^{3}+\frac{154432179539982}{24043381161601}a^{2}-\frac{27335144987143}{24043381161601}a-\frac{34609882270451}{24043381161601}$, $\frac{2940178876905}{24043381161601}a^{19}-\frac{13377417529132}{24043381161601}a^{18}+\frac{19974492911289}{24043381161601}a^{17}-\frac{70859451445111}{24043381161601}a^{16}+\frac{78430571165984}{24043381161601}a^{15}-\frac{122076573078545}{24043381161601}a^{14}+\frac{2170474083159}{24043381161601}a^{13}-\frac{17745741371313}{24043381161601}a^{12}-\frac{444064755770057}{24043381161601}a^{11}+\frac{435576250727490}{24043381161601}a^{10}-\frac{672066164177000}{24043381161601}a^{9}+\frac{538116605315262}{24043381161601}a^{8}-\frac{203198440916880}{24043381161601}a^{7}+\frac{129399393305783}{24043381161601}a^{6}+\frac{363296024277441}{24043381161601}a^{5}-\frac{350818707727982}{24043381161601}a^{4}+\frac{146681357371967}{24043381161601}a^{3}-\frac{209834112983385}{24043381161601}a^{2}+\frac{28094972701460}{24043381161601}a+\frac{27465816828364}{24043381161601}$, $\frac{6549517415011}{24043381161601}a^{19}-\frac{3093790808873}{24043381161601}a^{18}+\frac{41544352217128}{24043381161601}a^{17}-\frac{37324759764277}{24043381161601}a^{16}+\frac{123658986033470}{24043381161601}a^{15}-\frac{65589067721369}{24043381161601}a^{14}+\frac{217368850102592}{24043381161601}a^{13}+\frac{93803845550677}{24043381161601}a^{12}+\frac{11293937190469}{24043381161601}a^{11}+\frac{430811667841530}{24043381161601}a^{10}-\frac{585542256648682}{24043381161601}a^{9}+\frac{736625482522508}{24043381161601}a^{8}-\frac{983948239141671}{24043381161601}a^{7}+\frac{724355095532463}{24043381161601}a^{6}-\frac{706151767769133}{24043381161601}a^{5}+\frac{515583180583036}{24043381161601}a^{4}-\frac{157168326772578}{24043381161601}a^{3}+\frac{153173398339407}{24043381161601}a^{2}-\frac{27896485724351}{24043381161601}a-\frac{28982269521579}{24043381161601}$, $\frac{5203863801067}{24043381161601}a^{19}-\frac{1470170769304}{24043381161601}a^{18}+\frac{24580344510633}{24043381161601}a^{17}-\frac{16488895783409}{24043381161601}a^{16}+\frac{47884049480208}{24043381161601}a^{15}+\frac{18796350617883}{24043381161601}a^{14}+\frac{40449067554494}{24043381161601}a^{13}+\frac{160908631383423}{24043381161601}a^{12}-\frac{117779674722153}{24043381161601}a^{11}+\frac{182126195286355}{24043381161601}a^{10}-\frac{189927906002053}{24043381161601}a^{9}-\frac{7367382022670}{24043381161601}a^{8}-\frac{64534117700788}{24043381161601}a^{7}-\frac{129701476783862}{24043381161601}a^{6}+\frac{149447720514764}{24043381161601}a^{5}+\frac{25039060186039}{24043381161601}a^{4}+\frac{95905599416954}{24043381161601}a^{3}-\frac{20018870938431}{24043381161601}a^{2}-\frac{19121834196023}{24043381161601}a-\frac{10394033677089}{24043381161601}$, $\frac{3346743769916}{24043381161601}a^{19}+\frac{9218256962781}{24043381161601}a^{18}+\frac{7729681730078}{24043381161601}a^{17}+\frac{42394749815684}{24043381161601}a^{16}-\frac{29288368241340}{24043381161601}a^{15}+\frac{144763859571951}{24043381161601}a^{14}-\frac{11883147107693}{24043381161601}a^{13}+\frac{231078297263461}{24043381161601}a^{12}+\frac{155333699816739}{24043381161601}a^{11}-\frac{184396429299749}{24043381161601}a^{10}+\frac{415479437423703}{24043381161601}a^{9}-\frac{643178657332643}{24043381161601}a^{8}+\frac{417337687763311}{24043381161601}a^{7}-\frac{585559614622993}{24043381161601}a^{6}+\frac{253735742288237}{24043381161601}a^{5}+\frac{27292002571395}{24043381161601}a^{4}+\frac{144918828401044}{24043381161601}a^{3}+\frac{40013596260517}{24043381161601}a^{2}-\frac{70967322368560}{24043381161601}a-\frac{29142584452152}{24043381161601}$, $\frac{5135017373904}{24043381161601}a^{19}-\frac{4582086908412}{24043381161601}a^{18}+\frac{30057693188835}{24043381161601}a^{17}-\frac{33177970155449}{24043381161601}a^{16}+\frac{85406350261551}{24043381161601}a^{15}-\frac{31684997674456}{24043381161601}a^{14}+\frac{99841315420443}{24043381161601}a^{13}+\frac{125487805631856}{24043381161601}a^{12}-\frac{123603144939824}{24043381161601}a^{11}+\frac{399202875969878}{24043381161601}a^{10}-\frac{376099784150073}{24043381161601}a^{9}+\frac{417399209250469}{24043381161601}a^{8}-\frac{359037459210187}{24043381161601}a^{7}+\frac{128056620869517}{24043381161601}a^{6}-\frac{44815096035286}{24043381161601}a^{5}-\frac{68312132813110}{24043381161601}a^{4}+\frac{29625111262874}{24043381161601}a^{3}-\frac{27463264396664}{24043381161601}a^{2}+\frac{41494656398482}{24043381161601}a+\frac{12604552673074}{24043381161601}$, $\frac{8306381446980}{24043381161601}a^{19}-\frac{5241471556943}{24043381161601}a^{18}+\frac{35964983519220}{24043381161601}a^{17}-\frac{39018840370941}{24043381161601}a^{16}+\frac{67835914721062}{24043381161601}a^{15}+\frac{14812582300088}{24043381161601}a^{14}+\frac{21075511880175}{24043381161601}a^{13}+\frac{212667224618181}{24043381161601}a^{12}-\frac{307029172025482}{24043381161601}a^{11}+\frac{236400470388542}{24043381161601}a^{10}-\frac{322839501408985}{24043381161601}a^{9}-\frac{29757227365134}{24043381161601}a^{8}-\frac{995353410311}{24043381161601}a^{7}-\frac{193779011500119}{24043381161601}a^{6}+\frac{340868399728271}{24043381161601}a^{5}+\frac{23947503394457}{24043381161601}a^{4}+\frac{13673210225864}{24043381161601}a^{3}-\frac{56704503285354}{24043381161601}a^{2}-\frac{81456743562051}{24043381161601}a+\frac{32154504701134}{24043381161601}$, $\frac{2990334872713}{24043381161601}a^{19}+\frac{1950899601020}{24043381161601}a^{18}+\frac{22252974593433}{24043381161601}a^{17}-\frac{1314713271600}{24043381161601}a^{16}+\frac{67092510703761}{24043381161601}a^{15}-\frac{8482898963993}{24043381161601}a^{14}+\frac{156735450260971}{24043381161601}a^{13}+\frac{85005251201944}{24043381161601}a^{12}+\frac{171636124364276}{24043381161601}a^{11}+\frac{252207149985270}{24043381161601}a^{10}-\frac{184411009356047}{24043381161601}a^{9}+\frac{388530178617495}{24043381161601}a^{8}-\frac{616666089659189}{24043381161601}a^{7}+\frac{370001982071678}{24043381161601}a^{6}-\frac{590146629277909}{24043381161601}a^{5}+\frac{287750675132467}{24043381161601}a^{4}-\frac{48321554416015}{24043381161601}a^{3}+\frac{137047076300963}{24043381161601}a^{2}+\frac{67441867252243}{24043381161601}a-\frac{42925408488801}{24043381161601}$ 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:  \( 2780.31736012 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 2780.31736012 \cdot 1}{2\cdot\sqrt{109115950462157402855641}}\cr\approx \mathstrut & 0.163560784635 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*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 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*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 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*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 - x^19 + 6*x^18 - 8*x^17 + 18*x^16 - 13*x^15 + 25*x^14 + 10*x^13 - 23*x^12 + 65*x^11 - 103*x^10 + 107*x^9 - 130*x^8 + 94*x^7 - 65*x^6 + 53*x^5 - 12*x^4 + 5*x^3 - 3*x^2 - 4*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

$C_2^{10}:C_5$ (as 20T341):

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 5120
The 224 conjugacy class representatives for $C_2^{10}:C_5$
Character table for $C_2^{10}:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.14362045027.1, 10.6.330327035621.3, 10.4.4930254263.1

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
Minimal sibling: 20.4.109115950462157402855641.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(11\) Copy content Toggle raw display 11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$