Properties

Label 26.0.234...584.1
Degree $26$
Signature $[0, 13]$
Discriminant $-2.343\times 10^{53}$
Root discriminant \(112.90\)
Ramified primes $2,79$
Class number $189047$ (GRH)
Class group [189047] (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 85849)
 
gp: K = bnfinit(y^26 + 73*y^24 + 2180*y^22 + 35829*y^20 + 363625*y^18 + 2405673*y^16 + 10625163*y^14 + 31462021*y^12 + 61715876*y^10 + 77878153*y^8 + 59854953*y^6 + 25249803*y^4 + 4581570*y^2 + 85849, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 85849);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 85849)
 

\( x^{26} + 73 x^{24} + 2180 x^{22} + 35829 x^{20} + 363625 x^{18} + 2405673 x^{16} + 10625163 x^{14} + \cdots + 85849 \) 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:   \(-234331179334410135333725229627752024799305672723267584\) \(\medspace = -\,2^{26}\cdot 79^{24}\) 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:  \(112.90\)
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:  $2\cdot 79^{12/13}\approx 112.89787621344148$
Ramified primes:   \(2\), \(79\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(316=2^{2}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{316}(1,·)$, $\chi_{316}(259,·)$, $\chi_{316}(65,·)$, $\chi_{316}(141,·)$, $\chi_{316}(143,·)$, $\chi_{316}(275,·)$, $\chi_{316}(21,·)$, $\chi_{316}(87,·)$, $\chi_{316}(225,·)$, $\chi_{316}(89,·)$, $\chi_{316}(283,·)$, $\chi_{316}(159,·)$, $\chi_{316}(289,·)$, $\chi_{316}(67,·)$, $\chi_{316}(131,·)$, $\chi_{316}(101,·)$, $\chi_{316}(97,·)$, $\chi_{316}(299,·)$, $\chi_{316}(301,·)$, $\chi_{316}(255,·)$, $\chi_{316}(179,·)$, $\chi_{316}(245,·)$, $\chi_{316}(247,·)$, $\chi_{316}(223,·)$, $\chi_{316}(125,·)$, $\chi_{316}(117,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23}a^{16}+\frac{6}{23}a^{14}-\frac{2}{23}a^{12}-\frac{3}{23}a^{10}+\frac{8}{23}a^{8}+\frac{10}{23}a^{6}+\frac{11}{23}a^{4}-\frac{5}{23}a^{2}-\frac{10}{23}$, $\frac{1}{23}a^{17}+\frac{6}{23}a^{15}-\frac{2}{23}a^{13}-\frac{3}{23}a^{11}+\frac{8}{23}a^{9}+\frac{10}{23}a^{7}+\frac{11}{23}a^{5}-\frac{5}{23}a^{3}-\frac{10}{23}a$, $\frac{1}{23}a^{18}+\frac{8}{23}a^{14}+\frac{9}{23}a^{12}+\frac{3}{23}a^{10}+\frac{8}{23}a^{8}-\frac{3}{23}a^{6}-\frac{2}{23}a^{4}-\frac{3}{23}a^{2}-\frac{9}{23}$, $\frac{1}{23}a^{19}+\frac{8}{23}a^{15}+\frac{9}{23}a^{13}+\frac{3}{23}a^{11}+\frac{8}{23}a^{9}-\frac{3}{23}a^{7}-\frac{2}{23}a^{5}-\frac{3}{23}a^{3}-\frac{9}{23}a$, $\frac{1}{54487}a^{20}+\frac{969}{54487}a^{18}-\frac{706}{54487}a^{16}-\frac{15958}{54487}a^{14}+\frac{1757}{54487}a^{12}-\frac{17092}{54487}a^{10}+\frac{17125}{54487}a^{8}+\frac{23577}{54487}a^{6}+\frac{26660}{54487}a^{4}+\frac{18387}{54487}a^{2}+\frac{24846}{54487}$, $\frac{1}{54487}a^{21}+\frac{969}{54487}a^{19}-\frac{706}{54487}a^{17}-\frac{15958}{54487}a^{15}+\frac{1757}{54487}a^{13}-\frac{17092}{54487}a^{11}+\frac{17125}{54487}a^{9}+\frac{23577}{54487}a^{7}+\frac{26660}{54487}a^{5}+\frac{18387}{54487}a^{3}+\frac{24846}{54487}a$, $\frac{1}{1253201}a^{22}+\frac{11}{1253201}a^{20}-\frac{21681}{1253201}a^{18}+\frac{13653}{1253201}a^{16}+\frac{578031}{1253201}a^{14}-\frac{1414}{54487}a^{12}-\frac{468912}{1253201}a^{10}+\frac{577498}{1253201}a^{8}-\frac{582893}{1253201}a^{6}+\frac{94104}{1253201}a^{4}+\frac{248670}{1253201}a^{2}+\frac{434771}{1253201}$, $\frac{1}{1253201}a^{23}+\frac{11}{1253201}a^{21}-\frac{21681}{1253201}a^{19}+\frac{13653}{1253201}a^{17}+\frac{578031}{1253201}a^{15}-\frac{1414}{54487}a^{13}-\frac{468912}{1253201}a^{11}+\frac{577498}{1253201}a^{9}-\frac{582893}{1253201}a^{7}+\frac{94104}{1253201}a^{5}+\frac{248670}{1253201}a^{3}+\frac{434771}{1253201}a$, $\frac{1}{36\!\cdots\!91}a^{24}-\frac{3042854934540}{36\!\cdots\!91}a^{22}+\frac{163102868478168}{36\!\cdots\!91}a^{20}-\frac{72\!\cdots\!36}{36\!\cdots\!91}a^{18}-\frac{80\!\cdots\!98}{36\!\cdots\!91}a^{16}-\frac{96\!\cdots\!98}{36\!\cdots\!91}a^{14}-\frac{17\!\cdots\!79}{36\!\cdots\!91}a^{12}-\frac{14\!\cdots\!55}{36\!\cdots\!91}a^{10}+\frac{12\!\cdots\!26}{36\!\cdots\!91}a^{8}-\frac{74\!\cdots\!90}{36\!\cdots\!91}a^{6}+\frac{10\!\cdots\!92}{36\!\cdots\!91}a^{4}+\frac{53\!\cdots\!78}{15\!\cdots\!17}a^{2}-\frac{27\!\cdots\!50}{36\!\cdots\!91}$, $\frac{1}{10\!\cdots\!63}a^{25}-\frac{23\!\cdots\!20}{10\!\cdots\!63}a^{23}+\frac{65\!\cdots\!29}{10\!\cdots\!63}a^{21}-\frac{13\!\cdots\!19}{10\!\cdots\!63}a^{19}-\frac{96\!\cdots\!84}{10\!\cdots\!63}a^{17}-\frac{16\!\cdots\!38}{10\!\cdots\!63}a^{15}-\frac{44\!\cdots\!79}{10\!\cdots\!63}a^{13}-\frac{46\!\cdots\!06}{10\!\cdots\!63}a^{11}-\frac{64\!\cdots\!58}{10\!\cdots\!63}a^{9}-\frac{43\!\cdots\!66}{10\!\cdots\!63}a^{7}-\frac{51\!\cdots\!22}{10\!\cdots\!63}a^{5}+\frac{95\!\cdots\!38}{46\!\cdots\!81}a^{3}+\frac{22\!\cdots\!62}{10\!\cdots\!63}a$ 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

$C_{189047}$, which has order $189047$ (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:   \( \frac{1971930520834328}{462584552741787357181} a^{25} + \frac{140923162736181882}{462584552741787357181} a^{23} + \frac{4079661439500119245}{462584552741787357181} a^{21} + \frac{64194256870339635003}{462584552741787357181} a^{19} + \frac{612991798622570530525}{462584552741787357181} a^{17} + \frac{3718263039526908026100}{462584552741787357181} a^{15} + \frac{14451882346560990127351}{462584552741787357181} a^{13} + \frac{35086534019966094031366}{462584552741787357181} a^{11} + \frac{49061702742471158935382}{462584552741787357181} a^{9} + \frac{30259970969665189480089}{462584552741787357181} a^{7} - \frac{5415753680691374056530}{462584552741787357181} a^{5} - \frac{604858056457361772475}{20112371858338580747} a^{3} - \frac{3477616094881911388040}{462584552741787357181} a \)  (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:   $\frac{69\!\cdots\!09}{36\!\cdots\!91}a^{24}+\frac{48\!\cdots\!28}{36\!\cdots\!91}a^{22}+\frac{13\!\cdots\!72}{36\!\cdots\!91}a^{20}+\frac{21\!\cdots\!92}{36\!\cdots\!91}a^{18}+\frac{20\!\cdots\!06}{36\!\cdots\!91}a^{16}+\frac{12\!\cdots\!42}{36\!\cdots\!91}a^{14}+\frac{49\!\cdots\!01}{36\!\cdots\!91}a^{12}+\frac{12\!\cdots\!86}{36\!\cdots\!91}a^{10}+\frac{20\!\cdots\!02}{36\!\cdots\!91}a^{8}+\frac{20\!\cdots\!71}{36\!\cdots\!91}a^{6}+\frac{10\!\cdots\!26}{36\!\cdots\!91}a^{4}+\frac{10\!\cdots\!24}{15\!\cdots\!17}a^{2}+\frac{45\!\cdots\!75}{36\!\cdots\!91}$, $\frac{15\!\cdots\!81}{36\!\cdots\!91}a^{24}+\frac{11\!\cdots\!19}{36\!\cdots\!91}a^{22}+\frac{31\!\cdots\!19}{36\!\cdots\!91}a^{20}+\frac{47\!\cdots\!42}{36\!\cdots\!91}a^{18}+\frac{43\!\cdots\!31}{36\!\cdots\!91}a^{16}+\frac{25\!\cdots\!52}{36\!\cdots\!91}a^{14}+\frac{99\!\cdots\!99}{36\!\cdots\!91}a^{12}+\frac{24\!\cdots\!37}{36\!\cdots\!91}a^{10}+\frac{39\!\cdots\!06}{36\!\cdots\!91}a^{8}+\frac{39\!\cdots\!28}{36\!\cdots\!91}a^{6}+\frac{21\!\cdots\!98}{36\!\cdots\!91}a^{4}+\frac{21\!\cdots\!44}{15\!\cdots\!17}a^{2}+\frac{83\!\cdots\!65}{36\!\cdots\!91}$, $\frac{24\!\cdots\!99}{36\!\cdots\!91}a^{24}+\frac{16\!\cdots\!20}{36\!\cdots\!91}a^{22}+\frac{42\!\cdots\!32}{36\!\cdots\!91}a^{20}+\frac{57\!\cdots\!36}{36\!\cdots\!91}a^{18}+\frac{44\!\cdots\!60}{36\!\cdots\!91}a^{16}+\frac{19\!\cdots\!89}{36\!\cdots\!91}a^{14}+\frac{43\!\cdots\!27}{36\!\cdots\!91}a^{12}+\frac{10\!\cdots\!87}{36\!\cdots\!91}a^{10}-\frac{17\!\cdots\!26}{36\!\cdots\!91}a^{8}-\frac{36\!\cdots\!53}{36\!\cdots\!91}a^{6}-\frac{30\!\cdots\!11}{36\!\cdots\!91}a^{4}-\frac{39\!\cdots\!26}{15\!\cdots\!17}a^{2}-\frac{17\!\cdots\!25}{36\!\cdots\!91}$, $\frac{27\!\cdots\!82}{36\!\cdots\!91}a^{24}+\frac{19\!\cdots\!77}{36\!\cdots\!91}a^{22}+\frac{52\!\cdots\!10}{36\!\cdots\!91}a^{20}+\frac{79\!\cdots\!76}{36\!\cdots\!91}a^{18}+\frac{70\!\cdots\!78}{35\!\cdots\!97}a^{16}+\frac{41\!\cdots\!24}{36\!\cdots\!91}a^{14}+\frac{15\!\cdots\!16}{36\!\cdots\!91}a^{12}+\frac{38\!\cdots\!26}{36\!\cdots\!91}a^{10}+\frac{61\!\cdots\!01}{36\!\cdots\!91}a^{8}+\frac{59\!\cdots\!03}{36\!\cdots\!91}a^{6}+\frac{31\!\cdots\!51}{36\!\cdots\!91}a^{4}+\frac{32\!\cdots\!19}{15\!\cdots\!17}a^{2}+\frac{25\!\cdots\!16}{36\!\cdots\!91}$, $\frac{65\!\cdots\!84}{36\!\cdots\!91}a^{24}+\frac{47\!\cdots\!36}{36\!\cdots\!91}a^{22}+\frac{13\!\cdots\!54}{36\!\cdots\!91}a^{20}+\frac{21\!\cdots\!81}{36\!\cdots\!91}a^{18}+\frac{21\!\cdots\!77}{36\!\cdots\!91}a^{16}+\frac{13\!\cdots\!29}{36\!\cdots\!91}a^{14}+\frac{55\!\cdots\!19}{36\!\cdots\!91}a^{12}+\frac{14\!\cdots\!19}{35\!\cdots\!97}a^{10}+\frac{26\!\cdots\!62}{36\!\cdots\!91}a^{8}+\frac{27\!\cdots\!56}{36\!\cdots\!91}a^{6}+\frac{15\!\cdots\!23}{36\!\cdots\!91}a^{4}+\frac{16\!\cdots\!48}{15\!\cdots\!17}a^{2}+\frac{81\!\cdots\!63}{36\!\cdots\!91}$, $\frac{16\!\cdots\!53}{36\!\cdots\!91}a^{24}+\frac{11\!\cdots\!29}{36\!\cdots\!91}a^{22}+\frac{34\!\cdots\!28}{36\!\cdots\!91}a^{20}+\frac{53\!\cdots\!72}{36\!\cdots\!91}a^{18}+\frac{51\!\cdots\!46}{36\!\cdots\!91}a^{16}+\frac{31\!\cdots\!06}{36\!\cdots\!91}a^{14}+\frac{12\!\cdots\!27}{36\!\cdots\!91}a^{12}+\frac{33\!\cdots\!39}{36\!\cdots\!91}a^{10}+\frac{56\!\cdots\!18}{36\!\cdots\!91}a^{8}+\frac{57\!\cdots\!36}{36\!\cdots\!91}a^{6}+\frac{32\!\cdots\!77}{36\!\cdots\!91}a^{4}+\frac{32\!\cdots\!05}{15\!\cdots\!17}a^{2}+\frac{13\!\cdots\!97}{36\!\cdots\!91}$, $\frac{21\!\cdots\!33}{36\!\cdots\!91}a^{24}+\frac{14\!\cdots\!06}{36\!\cdots\!91}a^{22}+\frac{40\!\cdots\!73}{36\!\cdots\!91}a^{20}+\frac{59\!\cdots\!54}{36\!\cdots\!91}a^{18}+\frac{52\!\cdots\!72}{36\!\cdots\!91}a^{16}+\frac{28\!\cdots\!31}{36\!\cdots\!91}a^{14}+\frac{99\!\cdots\!16}{36\!\cdots\!91}a^{12}+\frac{21\!\cdots\!99}{36\!\cdots\!91}a^{10}+\frac{29\!\cdots\!87}{36\!\cdots\!91}a^{8}+\frac{22\!\cdots\!92}{36\!\cdots\!91}a^{6}+\frac{92\!\cdots\!12}{36\!\cdots\!91}a^{4}+\frac{65\!\cdots\!61}{15\!\cdots\!17}a^{2}+\frac{20\!\cdots\!68}{36\!\cdots\!91}$, $\frac{20\!\cdots\!78}{36\!\cdots\!91}a^{24}+\frac{14\!\cdots\!59}{36\!\cdots\!91}a^{22}+\frac{38\!\cdots\!22}{36\!\cdots\!91}a^{20}+\frac{56\!\cdots\!55}{36\!\cdots\!91}a^{18}+\frac{49\!\cdots\!19}{36\!\cdots\!91}a^{16}+\frac{26\!\cdots\!09}{36\!\cdots\!91}a^{14}+\frac{93\!\cdots\!41}{36\!\cdots\!91}a^{12}+\frac{20\!\cdots\!77}{36\!\cdots\!91}a^{10}+\frac{27\!\cdots\!95}{36\!\cdots\!91}a^{8}+\frac{22\!\cdots\!96}{36\!\cdots\!91}a^{6}+\frac{91\!\cdots\!41}{36\!\cdots\!91}a^{4}+\frac{68\!\cdots\!64}{15\!\cdots\!17}a^{2}+\frac{32\!\cdots\!18}{36\!\cdots\!91}$, $\frac{14\!\cdots\!51}{36\!\cdots\!91}a^{24}+\frac{10\!\cdots\!24}{36\!\cdots\!91}a^{22}+\frac{29\!\cdots\!11}{36\!\cdots\!91}a^{20}+\frac{46\!\cdots\!70}{36\!\cdots\!91}a^{18}+\frac{44\!\cdots\!77}{36\!\cdots\!91}a^{16}+\frac{27\!\cdots\!65}{36\!\cdots\!91}a^{14}+\frac{11\!\cdots\!21}{36\!\cdots\!91}a^{12}+\frac{30\!\cdots\!60}{36\!\cdots\!91}a^{10}+\frac{52\!\cdots\!93}{36\!\cdots\!91}a^{8}+\frac{55\!\cdots\!56}{36\!\cdots\!91}a^{6}+\frac{31\!\cdots\!08}{36\!\cdots\!91}a^{4}+\frac{33\!\cdots\!11}{15\!\cdots\!17}a^{2}+\frac{14\!\cdots\!87}{36\!\cdots\!91}$, $\frac{10\!\cdots\!11}{36\!\cdots\!91}a^{24}+\frac{76\!\cdots\!78}{36\!\cdots\!91}a^{22}+\frac{21\!\cdots\!07}{36\!\cdots\!91}a^{20}+\frac{34\!\cdots\!22}{36\!\cdots\!91}a^{18}+\frac{32\!\cdots\!65}{36\!\cdots\!91}a^{16}+\frac{20\!\cdots\!51}{36\!\cdots\!91}a^{14}+\frac{80\!\cdots\!61}{36\!\cdots\!91}a^{12}+\frac{21\!\cdots\!51}{36\!\cdots\!91}a^{10}+\frac{34\!\cdots\!83}{36\!\cdots\!91}a^{8}+\frac{34\!\cdots\!80}{36\!\cdots\!91}a^{6}+\frac{18\!\cdots\!12}{36\!\cdots\!91}a^{4}+\frac{18\!\cdots\!71}{15\!\cdots\!17}a^{2}+\frac{77\!\cdots\!69}{36\!\cdots\!91}$, $\frac{28\!\cdots\!72}{36\!\cdots\!91}a^{24}+\frac{19\!\cdots\!27}{36\!\cdots\!91}a^{22}+\frac{56\!\cdots\!59}{36\!\cdots\!91}a^{20}+\frac{84\!\cdots\!42}{36\!\cdots\!91}a^{18}+\frac{77\!\cdots\!80}{36\!\cdots\!91}a^{16}+\frac{45\!\cdots\!96}{36\!\cdots\!91}a^{14}+\frac{16\!\cdots\!66}{36\!\cdots\!91}a^{12}+\frac{40\!\cdots\!96}{36\!\cdots\!91}a^{10}+\frac{59\!\cdots\!79}{36\!\cdots\!91}a^{8}+\frac{50\!\cdots\!01}{36\!\cdots\!91}a^{6}+\frac{21\!\cdots\!61}{36\!\cdots\!91}a^{4}+\frac{14\!\cdots\!01}{15\!\cdots\!17}a^{2}-\frac{13\!\cdots\!79}{36\!\cdots\!91}$, $\frac{45\!\cdots\!56}{36\!\cdots\!91}a^{24}+\frac{31\!\cdots\!81}{36\!\cdots\!91}a^{22}+\frac{85\!\cdots\!97}{36\!\cdots\!91}a^{20}+\frac{12\!\cdots\!00}{36\!\cdots\!91}a^{18}+\frac{11\!\cdots\!24}{36\!\cdots\!91}a^{16}+\frac{63\!\cdots\!35}{36\!\cdots\!91}a^{14}+\frac{23\!\cdots\!10}{36\!\cdots\!91}a^{12}+\frac{53\!\cdots\!53}{36\!\cdots\!91}a^{10}+\frac{79\!\cdots\!99}{36\!\cdots\!91}a^{8}+\frac{70\!\cdots\!03}{36\!\cdots\!91}a^{6}+\frac{35\!\cdots\!59}{36\!\cdots\!91}a^{4}+\frac{35\!\cdots\!16}{15\!\cdots\!17}a^{2}+\frac{53\!\cdots\!51}{36\!\cdots\!91}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 189047}{4\cdot\sqrt{234331179334410135333725229627752024799305672723267584}}\cr\approx \mathstrut & 0.133606051949799 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 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 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 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 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 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 + 73*x^24 + 2180*x^22 + 35829*x^20 + 363625*x^18 + 2405673*x^16 + 10625163*x^14 + 31462021*x^12 + 61715876*x^10 + 77878153*x^8 + 59854953*x^6 + 25249803*x^4 + 4581570*x^2 + 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 $C_{26}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 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 R $26$ ${\href{/padicField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/13.13.0.1}{13} }^{2}$ ${\href{/padicField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/23.2.0.1}{2} }^{13}$ ${\href{/padicField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/37.13.0.1}{13} }^{2}$ ${\href{/padicField/41.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/53.13.0.1}{13} }^{2}$ $26$

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
\(2\) Copy content Toggle raw display Deg $26$$2$$13$$26$
\(79\) Copy content Toggle raw display Deg $26$$13$$2$$24$