Properties

Label 26.0.556...363.1
Degree $26$
Signature $[0, 13]$
Discriminant $-5.567\times 10^{51}$
Root discriminant \(97.77\)
Ramified primes $3,79$
Class number $57473$ (GRH)
Class group [57473] (GRH)
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849)
 
gp: K = bnfinit(y^26 - y^25 + 37*y^24 - 118*y^23 + 1008*y^22 - 3235*y^21 + 16642*y^20 - 51633*y^19 + 196206*y^18 - 526339*y^17 + 1528709*y^16 - 3470974*y^15 + 8121237*y^14 - 15101678*y^13 + 27024309*y^12 - 37166826*y^11 + 48531861*y^10 - 47702921*y^9 + 49613581*y^8 - 37639084*y^7 + 35389248*y^6 - 18788771*y^5 + 15390312*y^4 - 4606662*y^3 + 4701993*y^2 - 643428*y + 85849, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849)
 

\( x^{26} - x^{25} + 37 x^{24} - 118 x^{23} + 1008 x^{22} - 3235 x^{21} + 16642 x^{20} - 51633 x^{19} + 196206 x^{18} - 526339 x^{17} + 1528709 x^{16} - 3470974 x^{15} + \cdots + 85849 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-5567067695110660347277981181082226208360544116097363\) \(\medspace = -\,3^{13}\cdot 79^{24}\) 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:  \(97.77\)
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:  $3^{1/2}79^{12/13}\approx 97.77242883415123$
Ramified primes:   \(3\), \(79\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(237=3\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{237}(64,·)$, $\chi_{237}(1,·)$, $\chi_{237}(131,·)$, $\chi_{237}(196,·)$, $\chi_{237}(65,·)$, $\chi_{237}(8,·)$, $\chi_{237}(10,·)$, $\chi_{237}(143,·)$, $\chi_{237}(80,·)$, $\chi_{237}(146,·)$, $\chi_{237}(67,·)$, $\chi_{237}(22,·)$, $\chi_{237}(89,·)$, $\chi_{237}(220,·)$, $\chi_{237}(223,·)$, $\chi_{237}(97,·)$, $\chi_{237}(100,·)$, $\chi_{237}(101,·)$, $\chi_{237}(38,·)$, $\chi_{237}(46,·)$, $\chi_{237}(176,·)$, $\chi_{237}(179,·)$, $\chi_{237}(52,·)$, $\chi_{237}(125,·)$, $\chi_{237}(62,·)$, $\chi_{237}(166,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}-\frac{11}{23}a^{19}-\frac{7}{23}a^{18}+\frac{2}{23}a^{17}-\frac{11}{23}a^{16}+\frac{11}{23}a^{15}+\frac{6}{23}a^{14}-\frac{9}{23}a^{13}-\frac{10}{23}a^{12}-\frac{5}{23}a^{11}+\frac{10}{23}a^{10}+\frac{6}{23}a^{9}-\frac{9}{23}a^{8}-\frac{7}{23}a^{7}+\frac{11}{23}a^{6}+\frac{2}{23}a^{5}+\frac{6}{23}a^{4}-\frac{11}{23}a^{3}-\frac{5}{23}a^{2}-\frac{8}{23}a+\frac{6}{23}$, $\frac{1}{23}a^{21}+\frac{10}{23}a^{19}-\frac{6}{23}a^{18}+\frac{11}{23}a^{17}+\frac{5}{23}a^{16}-\frac{11}{23}a^{15}+\frac{11}{23}a^{14}+\frac{6}{23}a^{13}+\frac{1}{23}a^{11}+\frac{1}{23}a^{10}+\frac{11}{23}a^{9}+\frac{9}{23}a^{8}+\frac{3}{23}a^{7}+\frac{8}{23}a^{6}+\frac{5}{23}a^{5}+\frac{9}{23}a^{4}-\frac{11}{23}a^{3}+\frac{6}{23}a^{2}+\frac{10}{23}a-\frac{3}{23}$, $\frac{1}{23}a^{22}-\frac{11}{23}a^{19}-\frac{11}{23}a^{18}+\frac{8}{23}a^{17}+\frac{7}{23}a^{16}-\frac{7}{23}a^{15}-\frac{8}{23}a^{14}-\frac{2}{23}a^{13}+\frac{9}{23}a^{12}+\frac{5}{23}a^{11}+\frac{3}{23}a^{10}-\frac{5}{23}a^{9}+\frac{1}{23}a^{8}+\frac{9}{23}a^{7}+\frac{10}{23}a^{6}-\frac{11}{23}a^{5}-\frac{2}{23}a^{4}+\frac{1}{23}a^{3}-\frac{9}{23}a^{2}+\frac{8}{23}a+\frac{9}{23}$, $\frac{1}{428789}a^{23}-\frac{5069}{428789}a^{22}+\frac{8906}{428789}a^{21}-\frac{2061}{428789}a^{20}-\frac{117957}{428789}a^{19}-\frac{124539}{428789}a^{18}-\frac{35045}{428789}a^{17}+\frac{168026}{428789}a^{16}-\frac{102429}{428789}a^{15}-\frac{171887}{428789}a^{14}+\frac{157703}{428789}a^{13}+\frac{8136}{18643}a^{12}+\frac{204425}{428789}a^{11}+\frac{39296}{428789}a^{10}+\frac{91347}{428789}a^{9}+\frac{88070}{428789}a^{8}-\frac{179757}{428789}a^{7}-\frac{126778}{428789}a^{6}+\frac{190188}{428789}a^{5}-\frac{7532}{18643}a^{4}+\frac{26249}{428789}a^{3}+\frac{54161}{428789}a^{2}-\frac{176238}{428789}a+\frac{94462}{428789}$, $\frac{1}{226829381}a^{24}-\frac{172}{226829381}a^{23}+\frac{4213164}{226829381}a^{22}+\frac{4106104}{226829381}a^{21}-\frac{1858610}{226829381}a^{20}-\frac{47869979}{226829381}a^{19}+\frac{33616546}{226829381}a^{18}-\frac{4581589}{9862147}a^{17}+\frac{111770088}{226829381}a^{16}-\frac{1873298}{226829381}a^{15}+\frac{14662768}{226829381}a^{14}+\frac{74613943}{226829381}a^{13}-\frac{82731845}{226829381}a^{12}+\frac{58182867}{226829381}a^{11}+\frac{95505687}{226829381}a^{10}+\frac{90643238}{226829381}a^{9}-\frac{96218222}{226829381}a^{8}+\frac{86317315}{226829381}a^{7}+\frac{105801460}{226829381}a^{6}+\frac{31000145}{226829381}a^{5}+\frac{25263251}{226829381}a^{4}-\frac{90571594}{226829381}a^{3}-\frac{76881084}{226829381}a^{2}-\frac{109374321}{226829381}a+\frac{26296928}{226829381}$, $\frac{1}{49\!\cdots\!27}a^{25}+\frac{50\!\cdots\!64}{49\!\cdots\!27}a^{24}-\frac{54\!\cdots\!08}{49\!\cdots\!27}a^{23}+\frac{31\!\cdots\!48}{27\!\cdots\!67}a^{22}-\frac{89\!\cdots\!94}{49\!\cdots\!27}a^{21}+\frac{94\!\cdots\!00}{49\!\cdots\!27}a^{20}+\frac{18\!\cdots\!26}{49\!\cdots\!27}a^{19}-\frac{39\!\cdots\!38}{49\!\cdots\!27}a^{18}-\frac{14\!\cdots\!89}{49\!\cdots\!27}a^{17}-\frac{46\!\cdots\!76}{49\!\cdots\!27}a^{16}-\frac{77\!\cdots\!52}{21\!\cdots\!49}a^{15}-\frac{94\!\cdots\!55}{49\!\cdots\!27}a^{14}-\frac{17\!\cdots\!74}{49\!\cdots\!27}a^{13}+\frac{20\!\cdots\!43}{49\!\cdots\!27}a^{12}-\frac{46\!\cdots\!31}{21\!\cdots\!49}a^{11}-\frac{19\!\cdots\!17}{49\!\cdots\!27}a^{10}-\frac{11\!\cdots\!59}{49\!\cdots\!27}a^{9}+\frac{63\!\cdots\!70}{49\!\cdots\!27}a^{8}-\frac{36\!\cdots\!87}{49\!\cdots\!27}a^{7}+\frac{17\!\cdots\!97}{49\!\cdots\!27}a^{6}-\frac{20\!\cdots\!72}{49\!\cdots\!27}a^{5}+\frac{23\!\cdots\!00}{49\!\cdots\!27}a^{4}+\frac{12\!\cdots\!56}{49\!\cdots\!27}a^{3}-\frac{54\!\cdots\!66}{49\!\cdots\!27}a^{2}+\frac{19\!\cdots\!90}{49\!\cdots\!27}a+\frac{18\!\cdots\!62}{16\!\cdots\!39}$ 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

$C_{57473}$, which has order $57473$ (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:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{745437435626525585791097888709341304681484807006785254875549}{408484780404460089039416995905839193109216802389807767712954131881} a^{25} - \frac{800600851644091915550693456619893048232672919112354031509062}{408484780404460089039416995905839193109216802389807767712954131881} a^{24} + \frac{27633491585445404582125825140468867200147918128306953923210841}{408484780404460089039416995905839193109216802389807767712954131881} a^{23} - \frac{89932332821032303795495171336158580850423032047539225217894220}{408484780404460089039416995905839193109216802389807767712954131881} a^{22} + \frac{757736344994103711379521133711857773162434426200308472994013920}{408484780404460089039416995905839193109216802389807767712954131881} a^{21} - \frac{2464288908392365058113282363130840345871255333259680285988693784}{408484780404460089039416995905839193109216802389807767712954131881} a^{20} + \frac{12573204623928256474854921990373761147483149692735488756348734413}{408484780404460089039416995905839193109216802389807767712954131881} a^{19} - \frac{39330911361288242705889494852756800570596591189199619922913689537}{408484780404460089039416995905839193109216802389807767712954131881} a^{18} + \frac{148845080043023163759577860857899614387193792271760207280706212906}{408484780404460089039416995905839193109216802389807767712954131881} a^{17} - \frac{401933652602234683585264079828767046313057346437865059333368878931}{408484780404460089039416995905839193109216802389807767712954131881} a^{16} + \frac{1164644218294350376610152560579646099693911031526036638568843250948}{408484780404460089039416995905839193109216802389807767712954131881} a^{15} - \frac{2657455733490793958085602735923700346711360631808299696136786022297}{408484780404460089039416995905839193109216802389807767712954131881} a^{14} + \frac{6207029985588047906671405250161930715896574910644884226087144376576}{408484780404460089039416995905839193109216802389807767712954131881} a^{13} - \frac{11598064215401636134133795453151717443426255953600115214049227112301}{408484780404460089039416995905839193109216802389807767712954131881} a^{12} + \frac{20736838298104688526927598480458816125218529837050164805240642217353}{408484780404460089039416995905839193109216802389807767712954131881} a^{11} - \frac{28650958448430114553819787935661239455313835538399976808195141005716}{408484780404460089039416995905839193109216802389807767712954131881} a^{10} + \frac{37238537460543477723099146238779884229080542315073926124361974436810}{408484780404460089039416995905839193109216802389807767712954131881} a^{9} - \frac{36549407897392657125357430970714790943501781202717326421796785248527}{408484780404460089039416995905839193109216802389807767712954131881} a^{8} + \frac{37429046557079935182304712656079883099486240822683688346665522654115}{408484780404460089039416995905839193109216802389807767712954131881} a^{7} - \frac{28167755699276684561756024859865494506498191349179582440485315490428}{408484780404460089039416995905839193109216802389807767712954131881} a^{6} + \frac{26158145806174808843074478835580681763373827382739869584454294313980}{408484780404460089039416995905839193109216802389807767712954131881} a^{5} - \frac{13699740916400386158370648641827002546266938032398234891827527703172}{408484780404460089039416995905839193109216802389807767712954131881} a^{4} + \frac{10905024500702360702073152572174989223827794283152320479011396627999}{408484780404460089039416995905839193109216802389807767712954131881} a^{3} - \frac{2604055139933440516041199089854875479500442801564892131076838478333}{408484780404460089039416995905839193109216802389807767712954131881} a^{2} + \frac{3161861817590699445201304561941655417888284129745114459011791636626}{408484780404460089039416995905839193109216802389807767712954131881} a - \frac{78298508111038596112801991594433972609517842594379667909711265}{1394146008206348426755689405821976768290842328975453132126123317} \)  (order $6$) 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:   $\frac{52\!\cdots\!84}{49\!\cdots\!27}a^{25}-\frac{98\!\cdots\!42}{49\!\cdots\!27}a^{24}-\frac{89\!\cdots\!79}{49\!\cdots\!27}a^{23}-\frac{34\!\cdots\!06}{49\!\cdots\!27}a^{22}+\frac{53\!\cdots\!91}{49\!\cdots\!27}a^{21}-\frac{73\!\cdots\!09}{49\!\cdots\!27}a^{20}+\frac{14\!\cdots\!52}{49\!\cdots\!27}a^{19}-\frac{43\!\cdots\!85}{21\!\cdots\!49}a^{18}+\frac{23\!\cdots\!78}{49\!\cdots\!27}a^{17}-\frac{96\!\cdots\!38}{49\!\cdots\!27}a^{16}+\frac{19\!\cdots\!27}{49\!\cdots\!27}a^{15}-\frac{59\!\cdots\!77}{49\!\cdots\!27}a^{14}+\frac{10\!\cdots\!56}{49\!\cdots\!27}a^{13}-\frac{24\!\cdots\!12}{49\!\cdots\!27}a^{12}+\frac{34\!\cdots\!06}{49\!\cdots\!27}a^{11}-\frac{51\!\cdots\!17}{49\!\cdots\!27}a^{10}+\frac{42\!\cdots\!90}{49\!\cdots\!27}a^{9}-\frac{47\!\cdots\!87}{49\!\cdots\!27}a^{8}+\frac{26\!\cdots\!47}{49\!\cdots\!27}a^{7}-\frac{35\!\cdots\!61}{49\!\cdots\!27}a^{6}+\frac{12\!\cdots\!14}{49\!\cdots\!27}a^{5}-\frac{13\!\cdots\!18}{49\!\cdots\!27}a^{4}+\frac{90\!\cdots\!70}{49\!\cdots\!27}a^{3}-\frac{56\!\cdots\!37}{49\!\cdots\!27}a^{2}+\frac{35\!\cdots\!81}{49\!\cdots\!27}a-\frac{17\!\cdots\!11}{73\!\cdots\!93}$, $\frac{37\!\cdots\!49}{73\!\cdots\!93}a^{25}-\frac{11\!\cdots\!77}{16\!\cdots\!39}a^{24}+\frac{29\!\cdots\!71}{16\!\cdots\!39}a^{23}-\frac{11\!\cdots\!15}{16\!\cdots\!39}a^{22}+\frac{81\!\cdots\!00}{16\!\cdots\!39}a^{21}-\frac{29\!\cdots\!64}{16\!\cdots\!39}a^{20}+\frac{13\!\cdots\!83}{16\!\cdots\!39}a^{19}-\frac{44\!\cdots\!69}{16\!\cdots\!39}a^{18}+\frac{67\!\cdots\!41}{73\!\cdots\!93}a^{17}-\frac{43\!\cdots\!81}{16\!\cdots\!39}a^{16}+\frac{11\!\cdots\!01}{16\!\cdots\!39}a^{15}-\frac{26\!\cdots\!39}{16\!\cdots\!39}a^{14}+\frac{57\!\cdots\!56}{16\!\cdots\!39}a^{13}-\frac{10\!\cdots\!90}{16\!\cdots\!39}a^{12}+\frac{17\!\cdots\!51}{16\!\cdots\!39}a^{11}-\frac{22\!\cdots\!38}{16\!\cdots\!39}a^{10}+\frac{23\!\cdots\!54}{16\!\cdots\!39}a^{9}-\frac{21\!\cdots\!73}{16\!\cdots\!39}a^{8}+\frac{15\!\cdots\!06}{16\!\cdots\!39}a^{7}-\frac{13\!\cdots\!57}{16\!\cdots\!39}a^{6}+\frac{68\!\cdots\!59}{16\!\cdots\!39}a^{5}-\frac{43\!\cdots\!11}{16\!\cdots\!39}a^{4}-\frac{44\!\cdots\!41}{16\!\cdots\!39}a^{3}-\frac{76\!\cdots\!20}{16\!\cdots\!39}a^{2}+\frac{10\!\cdots\!10}{16\!\cdots\!39}a+\frac{89\!\cdots\!41}{16\!\cdots\!39}$, $\frac{64\!\cdots\!40}{73\!\cdots\!93}a^{25}-\frac{98\!\cdots\!55}{16\!\cdots\!39}a^{24}+\frac{55\!\cdots\!97}{16\!\cdots\!39}a^{23}-\frac{15\!\cdots\!49}{16\!\cdots\!39}a^{22}+\frac{14\!\cdots\!67}{16\!\cdots\!39}a^{21}-\frac{42\!\cdots\!60}{16\!\cdots\!39}a^{20}+\frac{23\!\cdots\!54}{16\!\cdots\!39}a^{19}-\frac{66\!\cdots\!60}{16\!\cdots\!39}a^{18}+\frac{11\!\cdots\!15}{73\!\cdots\!93}a^{17}-\frac{66\!\cdots\!90}{16\!\cdots\!39}a^{16}+\frac{19\!\cdots\!41}{16\!\cdots\!39}a^{15}-\frac{41\!\cdots\!07}{16\!\cdots\!39}a^{14}+\frac{95\!\cdots\!09}{16\!\cdots\!39}a^{13}-\frac{15\!\cdots\!25}{16\!\cdots\!39}a^{12}+\frac{27\!\cdots\!86}{16\!\cdots\!39}a^{11}-\frac{28\!\cdots\!12}{16\!\cdots\!39}a^{10}+\frac{27\!\cdots\!78}{16\!\cdots\!39}a^{9}-\frac{18\!\cdots\!85}{16\!\cdots\!39}a^{8}-\frac{87\!\cdots\!46}{16\!\cdots\!39}a^{7}+\frac{17\!\cdots\!74}{93\!\cdots\!19}a^{6}-\frac{19\!\cdots\!73}{16\!\cdots\!39}a^{5}+\frac{28\!\cdots\!32}{16\!\cdots\!39}a^{4}-\frac{13\!\cdots\!25}{16\!\cdots\!39}a^{3}+\frac{15\!\cdots\!46}{16\!\cdots\!39}a^{2}-\frac{22\!\cdots\!18}{16\!\cdots\!39}a+\frac{62\!\cdots\!82}{16\!\cdots\!39}$, $\frac{75\!\cdots\!22}{49\!\cdots\!27}a^{25}+\frac{53\!\cdots\!62}{49\!\cdots\!27}a^{24}+\frac{28\!\cdots\!14}{49\!\cdots\!27}a^{23}-\frac{44\!\cdots\!97}{49\!\cdots\!27}a^{22}+\frac{68\!\cdots\!94}{49\!\cdots\!27}a^{21}-\frac{14\!\cdots\!06}{49\!\cdots\!27}a^{20}+\frac{10\!\cdots\!36}{49\!\cdots\!27}a^{19}-\frac{10\!\cdots\!52}{21\!\cdots\!49}a^{18}+\frac{11\!\cdots\!02}{49\!\cdots\!27}a^{17}-\frac{25\!\cdots\!45}{49\!\cdots\!27}a^{16}+\frac{82\!\cdots\!80}{49\!\cdots\!27}a^{15}-\frac{16\!\cdots\!50}{49\!\cdots\!27}a^{14}+\frac{42\!\cdots\!91}{49\!\cdots\!27}a^{13}-\frac{70\!\cdots\!40}{49\!\cdots\!27}a^{12}+\frac{13\!\cdots\!82}{49\!\cdots\!27}a^{11}-\frac{16\!\cdots\!11}{49\!\cdots\!27}a^{10}+\frac{24\!\cdots\!64}{49\!\cdots\!27}a^{9}-\frac{21\!\cdots\!38}{49\!\cdots\!27}a^{8}+\frac{26\!\cdots\!40}{49\!\cdots\!27}a^{7}-\frac{15\!\cdots\!20}{49\!\cdots\!27}a^{6}+\frac{19\!\cdots\!96}{49\!\cdots\!27}a^{5}-\frac{68\!\cdots\!62}{49\!\cdots\!27}a^{4}+\frac{87\!\cdots\!92}{49\!\cdots\!27}a^{3}-\frac{80\!\cdots\!32}{49\!\cdots\!27}a^{2}+\frac{25\!\cdots\!39}{49\!\cdots\!27}a+\frac{12\!\cdots\!90}{73\!\cdots\!93}$, $\frac{12\!\cdots\!44}{49\!\cdots\!27}a^{25}+\frac{11\!\cdots\!96}{49\!\cdots\!27}a^{24}+\frac{44\!\cdots\!98}{49\!\cdots\!27}a^{23}-\frac{71\!\cdots\!87}{49\!\cdots\!27}a^{22}+\frac{97\!\cdots\!98}{49\!\cdots\!27}a^{21}-\frac{20\!\cdots\!07}{49\!\cdots\!27}a^{20}+\frac{13\!\cdots\!78}{49\!\cdots\!27}a^{19}-\frac{32\!\cdots\!52}{49\!\cdots\!27}a^{18}+\frac{13\!\cdots\!67}{49\!\cdots\!27}a^{17}-\frac{29\!\cdots\!17}{49\!\cdots\!27}a^{16}+\frac{86\!\cdots\!69}{49\!\cdots\!27}a^{15}-\frac{72\!\cdots\!75}{21\!\cdots\!49}a^{14}+\frac{20\!\cdots\!11}{27\!\cdots\!67}a^{13}-\frac{56\!\cdots\!46}{49\!\cdots\!27}a^{12}+\frac{88\!\cdots\!54}{49\!\cdots\!27}a^{11}-\frac{87\!\cdots\!04}{49\!\cdots\!27}a^{10}+\frac{99\!\cdots\!27}{49\!\cdots\!27}a^{9}-\frac{68\!\cdots\!92}{49\!\cdots\!27}a^{8}+\frac{68\!\cdots\!94}{49\!\cdots\!27}a^{7}-\frac{28\!\cdots\!90}{49\!\cdots\!27}a^{6}+\frac{17\!\cdots\!90}{49\!\cdots\!27}a^{5}+\frac{17\!\cdots\!43}{49\!\cdots\!27}a^{4}-\frac{27\!\cdots\!36}{49\!\cdots\!27}a^{3}+\frac{80\!\cdots\!33}{49\!\cdots\!27}a^{2}-\frac{70\!\cdots\!08}{49\!\cdots\!27}a+\frac{34\!\cdots\!55}{16\!\cdots\!39}$, $\frac{53\!\cdots\!41}{73\!\cdots\!93}a^{25}-\frac{14\!\cdots\!41}{16\!\cdots\!39}a^{24}+\frac{42\!\cdots\!83}{16\!\cdots\!39}a^{23}-\frac{15\!\cdots\!84}{16\!\cdots\!39}a^{22}+\frac{11\!\cdots\!69}{16\!\cdots\!39}a^{21}-\frac{39\!\cdots\!67}{16\!\cdots\!39}a^{20}+\frac{18\!\cdots\!00}{16\!\cdots\!39}a^{19}-\frac{60\!\cdots\!73}{16\!\cdots\!39}a^{18}+\frac{91\!\cdots\!65}{73\!\cdots\!93}a^{17}-\frac{58\!\cdots\!34}{16\!\cdots\!39}a^{16}+\frac{15\!\cdots\!44}{16\!\cdots\!39}a^{15}-\frac{35\!\cdots\!70}{16\!\cdots\!39}a^{14}+\frac{76\!\cdots\!69}{16\!\cdots\!39}a^{13}-\frac{14\!\cdots\!45}{16\!\cdots\!39}a^{12}+\frac{22\!\cdots\!90}{16\!\cdots\!39}a^{11}-\frac{27\!\cdots\!74}{16\!\cdots\!39}a^{10}+\frac{28\!\cdots\!98}{16\!\cdots\!39}a^{9}-\frac{24\!\cdots\!98}{16\!\cdots\!39}a^{8}+\frac{16\!\cdots\!26}{16\!\cdots\!39}a^{7}-\frac{11\!\cdots\!34}{16\!\cdots\!39}a^{6}+\frac{56\!\cdots\!50}{16\!\cdots\!39}a^{5}-\frac{18\!\cdots\!85}{16\!\cdots\!39}a^{4}-\frac{89\!\cdots\!04}{16\!\cdots\!39}a^{3}+\frac{80\!\cdots\!93}{16\!\cdots\!39}a^{2}-\frac{11\!\cdots\!93}{16\!\cdots\!39}a+\frac{79\!\cdots\!71}{16\!\cdots\!39}$, $\frac{47\!\cdots\!34}{20\!\cdots\!63}a^{25}-\frac{11\!\cdots\!94}{46\!\cdots\!49}a^{24}+\frac{35\!\cdots\!42}{46\!\cdots\!49}a^{23}-\frac{13\!\cdots\!56}{46\!\cdots\!49}a^{22}+\frac{94\!\cdots\!44}{46\!\cdots\!49}a^{21}-\frac{33\!\cdots\!80}{46\!\cdots\!49}a^{20}+\frac{14\!\cdots\!70}{46\!\cdots\!49}a^{19}-\frac{48\!\cdots\!35}{45\!\cdots\!83}a^{18}+\frac{72\!\cdots\!78}{20\!\cdots\!63}a^{17}-\frac{46\!\cdots\!00}{46\!\cdots\!49}a^{16}+\frac{11\!\cdots\!61}{46\!\cdots\!49}a^{15}-\frac{27\!\cdots\!34}{46\!\cdots\!49}a^{14}+\frac{57\!\cdots\!22}{46\!\cdots\!49}a^{13}-\frac{10\!\cdots\!02}{46\!\cdots\!49}a^{12}+\frac{15\!\cdots\!38}{46\!\cdots\!49}a^{11}-\frac{19\!\cdots\!46}{46\!\cdots\!49}a^{10}+\frac{19\!\cdots\!76}{46\!\cdots\!49}a^{9}-\frac{17\!\cdots\!24}{46\!\cdots\!49}a^{8}+\frac{13\!\cdots\!44}{46\!\cdots\!49}a^{7}-\frac{10\!\cdots\!67}{45\!\cdots\!83}a^{6}+\frac{52\!\cdots\!34}{46\!\cdots\!49}a^{5}-\frac{27\!\cdots\!38}{46\!\cdots\!49}a^{4}+\frac{30\!\cdots\!61}{46\!\cdots\!49}a^{3}-\frac{19\!\cdots\!96}{46\!\cdots\!49}a^{2}+\frac{22\!\cdots\!58}{46\!\cdots\!49}a+\frac{47\!\cdots\!04}{46\!\cdots\!49}$, $\frac{16\!\cdots\!94}{49\!\cdots\!27}a^{25}+\frac{22\!\cdots\!43}{49\!\cdots\!27}a^{24}+\frac{83\!\cdots\!66}{49\!\cdots\!27}a^{23}+\frac{64\!\cdots\!34}{49\!\cdots\!27}a^{22}+\frac{32\!\cdots\!92}{49\!\cdots\!27}a^{21}+\frac{13\!\cdots\!06}{49\!\cdots\!27}a^{20}-\frac{10\!\cdots\!52}{49\!\cdots\!27}a^{19}+\frac{73\!\cdots\!88}{21\!\cdots\!49}a^{18}-\frac{28\!\cdots\!50}{49\!\cdots\!27}a^{17}+\frac{16\!\cdots\!23}{49\!\cdots\!27}a^{16}-\frac{28\!\cdots\!83}{49\!\cdots\!27}a^{15}+\frac{10\!\cdots\!82}{49\!\cdots\!27}a^{14}-\frac{17\!\cdots\!47}{49\!\cdots\!27}a^{13}+\frac{45\!\cdots\!01}{49\!\cdots\!27}a^{12}-\frac{58\!\cdots\!57}{49\!\cdots\!27}a^{11}+\frac{10\!\cdots\!12}{49\!\cdots\!27}a^{10}-\frac{78\!\cdots\!73}{49\!\cdots\!27}a^{9}+\frac{10\!\cdots\!83}{49\!\cdots\!27}a^{8}-\frac{48\!\cdots\!26}{49\!\cdots\!27}a^{7}+\frac{86\!\cdots\!03}{49\!\cdots\!27}a^{6}-\frac{18\!\cdots\!72}{49\!\cdots\!27}a^{5}+\frac{44\!\cdots\!59}{49\!\cdots\!27}a^{4}+\frac{47\!\cdots\!77}{49\!\cdots\!27}a^{3}+\frac{16\!\cdots\!27}{49\!\cdots\!27}a^{2}+\frac{44\!\cdots\!51}{49\!\cdots\!27}a+\frac{55\!\cdots\!65}{73\!\cdots\!93}$, $\frac{67\!\cdots\!43}{73\!\cdots\!93}a^{25}-\frac{23\!\cdots\!90}{16\!\cdots\!39}a^{24}+\frac{54\!\cdots\!87}{16\!\cdots\!39}a^{23}-\frac{21\!\cdots\!97}{16\!\cdots\!39}a^{22}+\frac{15\!\cdots\!23}{16\!\cdots\!39}a^{21}-\frac{54\!\cdots\!20}{16\!\cdots\!39}a^{20}+\frac{24\!\cdots\!41}{16\!\cdots\!39}a^{19}-\frac{83\!\cdots\!66}{16\!\cdots\!39}a^{18}+\frac{12\!\cdots\!69}{73\!\cdots\!93}a^{17}-\frac{82\!\cdots\!17}{16\!\cdots\!39}a^{16}+\frac{22\!\cdots\!33}{16\!\cdots\!39}a^{15}-\frac{50\!\cdots\!01}{16\!\cdots\!13}a^{14}+\frac{11\!\cdots\!96}{16\!\cdots\!39}a^{13}-\frac{21\!\cdots\!63}{16\!\cdots\!39}a^{12}+\frac{33\!\cdots\!45}{16\!\cdots\!39}a^{11}-\frac{45\!\cdots\!72}{16\!\cdots\!39}a^{10}+\frac{48\!\cdots\!55}{16\!\cdots\!39}a^{9}-\frac{44\!\cdots\!43}{16\!\cdots\!39}a^{8}+\frac{33\!\cdots\!70}{16\!\cdots\!39}a^{7}-\frac{27\!\cdots\!65}{16\!\cdots\!39}a^{6}+\frac{15\!\cdots\!63}{16\!\cdots\!39}a^{5}-\frac{10\!\cdots\!37}{16\!\cdots\!39}a^{4}+\frac{13\!\cdots\!18}{16\!\cdots\!39}a^{3}-\frac{24\!\cdots\!08}{16\!\cdots\!39}a^{2}+\frac{32\!\cdots\!96}{16\!\cdots\!39}a-\frac{19\!\cdots\!70}{16\!\cdots\!39}$, $\frac{53\!\cdots\!53}{73\!\cdots\!93}a^{25}-\frac{11\!\cdots\!29}{16\!\cdots\!39}a^{24}+\frac{39\!\cdots\!76}{16\!\cdots\!39}a^{23}-\frac{14\!\cdots\!76}{16\!\cdots\!39}a^{22}+\frac{10\!\cdots\!24}{16\!\cdots\!39}a^{21}-\frac{35\!\cdots\!09}{16\!\cdots\!39}a^{20}+\frac{15\!\cdots\!04}{16\!\cdots\!39}a^{19}-\frac{53\!\cdots\!72}{16\!\cdots\!39}a^{18}+\frac{77\!\cdots\!22}{73\!\cdots\!93}a^{17}-\frac{49\!\cdots\!03}{16\!\cdots\!39}a^{16}+\frac{12\!\cdots\!48}{16\!\cdots\!39}a^{15}-\frac{29\!\cdots\!83}{16\!\cdots\!39}a^{14}+\frac{59\!\cdots\!65}{16\!\cdots\!39}a^{13}-\frac{10\!\cdots\!28}{16\!\cdots\!39}a^{12}+\frac{15\!\cdots\!18}{16\!\cdots\!39}a^{11}-\frac{19\!\cdots\!89}{16\!\cdots\!39}a^{10}+\frac{19\!\cdots\!66}{16\!\cdots\!39}a^{9}-\frac{17\!\cdots\!42}{16\!\cdots\!39}a^{8}+\frac{12\!\cdots\!12}{16\!\cdots\!39}a^{7}-\frac{94\!\cdots\!51}{16\!\cdots\!39}a^{6}+\frac{47\!\cdots\!13}{16\!\cdots\!39}a^{5}-\frac{20\!\cdots\!85}{16\!\cdots\!39}a^{4}+\frac{87\!\cdots\!88}{16\!\cdots\!39}a^{3}+\frac{13\!\cdots\!93}{16\!\cdots\!39}a^{2}-\frac{23\!\cdots\!92}{16\!\cdots\!39}a+\frac{86\!\cdots\!94}{16\!\cdots\!39}$, $\frac{18\!\cdots\!06}{49\!\cdots\!27}a^{25}+\frac{11\!\cdots\!38}{49\!\cdots\!27}a^{24}+\frac{63\!\cdots\!72}{49\!\cdots\!27}a^{23}-\frac{11\!\cdots\!50}{49\!\cdots\!27}a^{22}+\frac{14\!\cdots\!34}{49\!\cdots\!27}a^{21}-\frac{32\!\cdots\!94}{49\!\cdots\!27}a^{20}+\frac{20\!\cdots\!58}{49\!\cdots\!27}a^{19}-\frac{51\!\cdots\!60}{49\!\cdots\!27}a^{18}+\frac{20\!\cdots\!06}{49\!\cdots\!27}a^{17}-\frac{46\!\cdots\!84}{49\!\cdots\!27}a^{16}+\frac{13\!\cdots\!94}{49\!\cdots\!27}a^{15}-\frac{11\!\cdots\!57}{21\!\cdots\!49}a^{14}+\frac{61\!\cdots\!10}{49\!\cdots\!27}a^{13}-\frac{94\!\cdots\!58}{49\!\cdots\!27}a^{12}+\frac{15\!\cdots\!72}{49\!\cdots\!27}a^{11}-\frac{15\!\cdots\!62}{49\!\cdots\!27}a^{10}+\frac{17\!\cdots\!68}{49\!\cdots\!27}a^{9}-\frac{12\!\cdots\!00}{49\!\cdots\!27}a^{8}+\frac{14\!\cdots\!04}{49\!\cdots\!27}a^{7}-\frac{81\!\cdots\!32}{49\!\cdots\!27}a^{6}+\frac{75\!\cdots\!00}{49\!\cdots\!27}a^{5}-\frac{24\!\cdots\!34}{49\!\cdots\!27}a^{4}+\frac{21\!\cdots\!80}{49\!\cdots\!27}a^{3}-\frac{12\!\cdots\!72}{49\!\cdots\!27}a^{2}+\frac{10\!\cdots\!52}{49\!\cdots\!27}a-\frac{33\!\cdots\!32}{16\!\cdots\!39}$, $\frac{24\!\cdots\!16}{49\!\cdots\!27}a^{25}+\frac{66\!\cdots\!56}{49\!\cdots\!27}a^{24}+\frac{20\!\cdots\!66}{49\!\cdots\!27}a^{23}+\frac{20\!\cdots\!30}{49\!\cdots\!27}a^{22}+\frac{35\!\cdots\!36}{49\!\cdots\!27}a^{21}+\frac{37\!\cdots\!72}{49\!\cdots\!27}a^{20}-\frac{21\!\cdots\!80}{49\!\cdots\!27}a^{19}+\frac{19\!\cdots\!84}{21\!\cdots\!49}a^{18}-\frac{52\!\cdots\!74}{49\!\cdots\!27}a^{17}+\frac{19\!\cdots\!94}{27\!\cdots\!67}a^{16}-\frac{33\!\cdots\!04}{49\!\cdots\!27}a^{15}+\frac{14\!\cdots\!06}{49\!\cdots\!27}a^{14}-\frac{66\!\cdots\!52}{49\!\cdots\!27}a^{13}+\frac{21\!\cdots\!88}{49\!\cdots\!27}a^{12}+\frac{74\!\cdots\!42}{49\!\cdots\!27}a^{11}-\frac{18\!\cdots\!00}{49\!\cdots\!27}a^{10}+\frac{54\!\cdots\!18}{49\!\cdots\!27}a^{9}-\frac{71\!\cdots\!00}{49\!\cdots\!27}a^{8}+\frac{92\!\cdots\!70}{49\!\cdots\!27}a^{7}-\frac{68\!\cdots\!48}{49\!\cdots\!27}a^{6}+\frac{60\!\cdots\!88}{49\!\cdots\!27}a^{5}-\frac{34\!\cdots\!65}{48\!\cdots\!09}a^{4}+\frac{26\!\cdots\!48}{49\!\cdots\!27}a^{3}-\frac{87\!\cdots\!30}{49\!\cdots\!27}a^{2}+\frac{14\!\cdots\!82}{49\!\cdots\!27}a-\frac{15\!\cdots\!48}{73\!\cdots\!93}$ 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:  \( 57529828940.82975 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 57473}{6\cdot\sqrt{5567067695110660347277981181082226208360544116097363}}\cr\approx \mathstrut & 0.175683403748937 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849)
 
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^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849, 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^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849);
 
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^26 - x^25 + 37*x^24 - 118*x^23 + 1008*x^22 - 3235*x^21 + 16642*x^20 - 51633*x^19 + 196206*x^18 - 526339*x^17 + 1528709*x^16 - 3470974*x^15 + 8121237*x^14 - 15101678*x^13 + 27024309*x^12 - 37166826*x^11 + 48531861*x^10 - 47702921*x^9 + 49613581*x^8 - 37639084*x^7 + 35389248*x^6 - 18788771*x^5 + 15390312*x^4 - 4606662*x^3 + 4701993*x^2 - 643428*x + 85849);
 
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

$C_{26}$ (as 26T1):

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 cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 13.13.59091511031674153381441.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ R $26$ ${\href{/padicField/7.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/19.13.0.1}{13} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{13}$ $26$ ${\href{/padicField/31.13.0.1}{13} }^{2}$ ${\href{/padicField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/43.13.0.1}{13} }^{2}$ $26$ $26$ $26$

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
\(3\) Copy content Toggle raw display Deg $26$$2$$13$$13$
\(79\) Copy content Toggle raw display 79.13.12.1$x^{13} + 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} + 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$