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Properties

Label 40.0.890...496.1
Degree $40$
Signature $[0, 20]$
Discriminant $8.900\times 10^{66}$
Root discriminant \(47.18\)
Ramified primes $2,3,11$
Class number $125$ (GRH)
Class group [5, 5, 5] (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 1)
 
gp: K = bnfinit(y^40 - 25*y^36 + 441*y^32 - 3794*y^28 + 23626*y^24 - 66403*y^20 + 133864*y^16 - 62097*y^12 + 23622*y^8 - 155*y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 1)
 

\( x^{40} - 25 x^{36} + 441 x^{32} - 3794 x^{28} + 23626 x^{24} - 66403 x^{20} + 133864 x^{16} - 62097 x^{12} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(8900013646919506378267907122148784192909845412083315599549447274496\) \(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{32}\) 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:  \(47.18\)
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:  $2^{2}3^{1/2}11^{4/5}\approx 47.177483000606586$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(133,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(157,·)$, $\chi_{264}(5,·)$, $\chi_{264}(163,·)$, $\chi_{264}(37,·)$, $\chi_{264}(169,·)$, $\chi_{264}(71,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(181,·)$, $\chi_{264}(185,·)$, $\chi_{264}(31,·)$, $\chi_{264}(53,·)$, $\chi_{264}(67,·)$, $\chi_{264}(199,·)$, $\chi_{264}(203,·)$, $\chi_{264}(89,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(59,·)$, $\chi_{264}(229,·)$, $\chi_{264}(103,·)$, $\chi_{264}(235,·)$, $\chi_{264}(191,·)$, $\chi_{264}(113,·)$, $\chi_{264}(115,·)$, $\chi_{264}(245,·)$, $\chi_{264}(119,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(247,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{89893}a^{32}+\frac{14059}{89893}a^{28}+\frac{7913}{89893}a^{24}-\frac{39223}{89893}a^{20}-\frac{10045}{89893}a^{16}+\frac{32533}{89893}a^{12}+\frac{42331}{89893}a^{8}+\frac{10410}{89893}a^{4}+\frac{32919}{89893}$, $\frac{1}{89893}a^{33}+\frac{14059}{89893}a^{29}+\frac{7913}{89893}a^{25}-\frac{39223}{89893}a^{21}-\frac{10045}{89893}a^{17}+\frac{32533}{89893}a^{13}+\frac{42331}{89893}a^{9}+\frac{10410}{89893}a^{5}+\frac{32919}{89893}a$, $\frac{1}{89893}a^{34}+\frac{14059}{89893}a^{30}+\frac{7913}{89893}a^{26}-\frac{39223}{89893}a^{22}-\frac{10045}{89893}a^{18}+\frac{32533}{89893}a^{14}+\frac{42331}{89893}a^{10}+\frac{10410}{89893}a^{6}+\frac{32919}{89893}a^{2}$, $\frac{1}{89893}a^{35}+\frac{14059}{89893}a^{31}+\frac{7913}{89893}a^{27}-\frac{39223}{89893}a^{23}-\frac{10045}{89893}a^{19}+\frac{32533}{89893}a^{15}+\frac{42331}{89893}a^{11}+\frac{10410}{89893}a^{7}+\frac{32919}{89893}a^{3}$, $\frac{1}{22\!\cdots\!43}a^{36}-\frac{47\!\cdots\!37}{22\!\cdots\!43}a^{32}-\frac{15\!\cdots\!24}{22\!\cdots\!43}a^{28}+\frac{36\!\cdots\!98}{22\!\cdots\!43}a^{24}-\frac{11\!\cdots\!64}{22\!\cdots\!43}a^{20}+\frac{50\!\cdots\!55}{22\!\cdots\!43}a^{16}+\frac{31\!\cdots\!83}{22\!\cdots\!43}a^{12}+\frac{75\!\cdots\!71}{22\!\cdots\!43}a^{8}-\frac{11\!\cdots\!69}{22\!\cdots\!43}a^{4}-\frac{11\!\cdots\!70}{22\!\cdots\!43}$, $\frac{1}{22\!\cdots\!43}a^{37}-\frac{47\!\cdots\!37}{22\!\cdots\!43}a^{33}-\frac{15\!\cdots\!24}{22\!\cdots\!43}a^{29}+\frac{36\!\cdots\!98}{22\!\cdots\!43}a^{25}-\frac{11\!\cdots\!64}{22\!\cdots\!43}a^{21}+\frac{50\!\cdots\!55}{22\!\cdots\!43}a^{17}+\frac{31\!\cdots\!83}{22\!\cdots\!43}a^{13}+\frac{75\!\cdots\!71}{22\!\cdots\!43}a^{9}-\frac{11\!\cdots\!69}{22\!\cdots\!43}a^{5}-\frac{11\!\cdots\!70}{22\!\cdots\!43}a$, $\frac{1}{22\!\cdots\!43}a^{38}-\frac{47\!\cdots\!37}{22\!\cdots\!43}a^{34}-\frac{15\!\cdots\!24}{22\!\cdots\!43}a^{30}+\frac{36\!\cdots\!98}{22\!\cdots\!43}a^{26}-\frac{11\!\cdots\!64}{22\!\cdots\!43}a^{22}+\frac{50\!\cdots\!55}{22\!\cdots\!43}a^{18}+\frac{31\!\cdots\!83}{22\!\cdots\!43}a^{14}+\frac{75\!\cdots\!71}{22\!\cdots\!43}a^{10}-\frac{11\!\cdots\!69}{22\!\cdots\!43}a^{6}-\frac{11\!\cdots\!70}{22\!\cdots\!43}a^{2}$, $\frac{1}{22\!\cdots\!43}a^{39}-\frac{47\!\cdots\!37}{22\!\cdots\!43}a^{35}-\frac{15\!\cdots\!24}{22\!\cdots\!43}a^{31}+\frac{36\!\cdots\!98}{22\!\cdots\!43}a^{27}-\frac{11\!\cdots\!64}{22\!\cdots\!43}a^{23}+\frac{50\!\cdots\!55}{22\!\cdots\!43}a^{19}+\frac{31\!\cdots\!83}{22\!\cdots\!43}a^{15}+\frac{75\!\cdots\!71}{22\!\cdots\!43}a^{11}-\frac{11\!\cdots\!69}{22\!\cdots\!43}a^{7}-\frac{11\!\cdots\!70}{22\!\cdots\!43}a^{3}$ 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

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{53581368526244142240}{2273551084786794990343} a^{37} + \frac{1339224586826730664500}{2273551084786794990343} a^{33} - \frac{23621564515385087475643}{2273551084786794990343} a^{29} + \frac{203149221171675960743220}{2273551084786794990343} a^{25} - \frac{1264704590067397349415492}{2273551084786794990343} a^{21} + \frac{3550359893243219213346364}{2273551084786794990343} a^{17} - \frac{7150303866121008375115512}{2273551084786794990343} a^{13} + \frac{3281262936679115684907084}{2273551084786794990343} a^{9} - \frac{1238571944153848101248881}{2273551084786794990343} a^{5} + \frac{139324111753014574752}{2273551084786794990343} a \)  (order $24$) 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:   $\frac{28\!\cdots\!61}{22\!\cdots\!43}a^{38}-\frac{69\!\cdots\!80}{22\!\cdots\!43}a^{34}+\frac{12\!\cdots\!66}{22\!\cdots\!43}a^{30}-\frac{10\!\cdots\!14}{22\!\cdots\!43}a^{26}+\frac{62\!\cdots\!66}{22\!\cdots\!43}a^{22}-\frac{16\!\cdots\!82}{22\!\cdots\!43}a^{18}+\frac{12\!\cdots\!06}{94\!\cdots\!23}a^{14}-\frac{30\!\cdots\!16}{22\!\cdots\!43}a^{10}+\frac{19\!\cdots\!30}{22\!\cdots\!43}a^{6}+\frac{28\!\cdots\!63}{22\!\cdots\!43}a^{2}-1$, $\frac{336875892442205}{17\!\cdots\!71}a^{36}-\frac{84\!\cdots\!99}{17\!\cdots\!71}a^{32}+\frac{14\!\cdots\!27}{17\!\cdots\!71}a^{28}-\frac{12\!\cdots\!04}{17\!\cdots\!71}a^{24}+\frac{79\!\cdots\!01}{17\!\cdots\!71}a^{20}-\frac{22\!\cdots\!97}{17\!\cdots\!71}a^{16}+\frac{44\!\cdots\!57}{17\!\cdots\!71}a^{12}-\frac{20\!\cdots\!76}{17\!\cdots\!71}a^{8}+\frac{75\!\cdots\!63}{17\!\cdots\!71}a^{4}-\frac{876625318395493}{17\!\cdots\!71}$, $\frac{78\!\cdots\!87}{22\!\cdots\!43}a^{36}-\frac{19\!\cdots\!52}{22\!\cdots\!43}a^{32}+\frac{34\!\cdots\!47}{22\!\cdots\!43}a^{28}-\frac{29\!\cdots\!84}{22\!\cdots\!43}a^{24}+\frac{18\!\cdots\!32}{22\!\cdots\!43}a^{20}-\frac{51\!\cdots\!67}{22\!\cdots\!43}a^{16}+\frac{10\!\cdots\!30}{22\!\cdots\!43}a^{12}-\frac{47\!\cdots\!40}{22\!\cdots\!43}a^{8}+\frac{18\!\cdots\!70}{22\!\cdots\!43}a^{4}-\frac{12\!\cdots\!15}{22\!\cdots\!43}$, $\frac{48\!\cdots\!88}{22\!\cdots\!43}a^{36}-\frac{12\!\cdots\!13}{22\!\cdots\!43}a^{32}+\frac{21\!\cdots\!88}{22\!\cdots\!43}a^{28}-\frac{18\!\cdots\!43}{22\!\cdots\!43}a^{24}+\frac{11\!\cdots\!95}{22\!\cdots\!43}a^{20}-\frac{31\!\cdots\!35}{22\!\cdots\!43}a^{16}+\frac{64\!\cdots\!99}{22\!\cdots\!43}a^{12}-\frac{29\!\cdots\!12}{22\!\cdots\!43}a^{8}+\frac{11\!\cdots\!82}{22\!\cdots\!43}a^{4}-\frac{74\!\cdots\!45}{22\!\cdots\!43}$, $\frac{19\!\cdots\!08}{22\!\cdots\!43}a^{36}-\frac{48\!\cdots\!24}{22\!\cdots\!43}a^{32}+\frac{85\!\cdots\!28}{22\!\cdots\!43}a^{28}-\frac{73\!\cdots\!87}{22\!\cdots\!43}a^{24}+\frac{46\!\cdots\!74}{22\!\cdots\!43}a^{20}-\frac{12\!\cdots\!59}{22\!\cdots\!43}a^{16}+\frac{26\!\cdots\!07}{22\!\cdots\!43}a^{12}-\frac{12\!\cdots\!66}{22\!\cdots\!43}a^{8}+\frac{46\!\cdots\!86}{22\!\cdots\!43}a^{4}-\frac{30\!\cdots\!65}{22\!\cdots\!43}$, $\frac{12\!\cdots\!62}{22\!\cdots\!43}a^{37}-\frac{30\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{52\!\cdots\!45}{22\!\cdots\!43}a^{29}-\frac{44\!\cdots\!23}{22\!\cdots\!43}a^{25}+\frac{27\!\cdots\!45}{22\!\cdots\!43}a^{21}-\frac{70\!\cdots\!65}{22\!\cdots\!43}a^{17}+\frac{54\!\cdots\!15}{94\!\cdots\!23}a^{13}-\frac{13\!\cdots\!70}{22\!\cdots\!43}a^{9}+\frac{86\!\cdots\!75}{22\!\cdots\!43}a^{5}+\frac{79\!\cdots\!13}{22\!\cdots\!43}a+1$, $\frac{64\!\cdots\!51}{22\!\cdots\!43}a^{39}-\frac{106777091540}{1297652289253}a^{38}-\frac{16\!\cdots\!50}{22\!\cdots\!43}a^{35}+\frac{2668717678430}{1297652289253}a^{34}+\frac{28\!\cdots\!71}{22\!\cdots\!43}a^{31}-\frac{47071002697755}{1297652289253}a^{30}-\frac{24\!\cdots\!75}{22\!\cdots\!43}a^{27}+\frac{404800465622850}{1297652289253}a^{26}+\frac{15\!\cdots\!04}{22\!\cdots\!43}a^{23}-\frac{25\!\cdots\!22}{1297652289253}a^{22}-\frac{42\!\cdots\!34}{22\!\cdots\!43}a^{19}+\frac{70\!\cdots\!59}{1297652289253}a^{18}+\frac{86\!\cdots\!01}{22\!\cdots\!43}a^{15}-\frac{14\!\cdots\!12}{1297652289253}a^{14}-\frac{40\!\cdots\!30}{22\!\cdots\!43}a^{11}+\frac{65\!\cdots\!74}{1297652289253}a^{10}+\frac{15\!\cdots\!28}{22\!\cdots\!43}a^{7}-\frac{24\!\cdots\!88}{1297652289253}a^{6}-\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}+\frac{277613222772}{1297652289253}a^{2}$, $\frac{106777091540}{1297652289253}a^{38}+\frac{68\!\cdots\!88}{22\!\cdots\!43}a^{37}-\frac{2668717678430}{1297652289253}a^{34}-\frac{17\!\cdots\!25}{22\!\cdots\!43}a^{33}+\frac{47071002697755}{1297652289253}a^{30}+\frac{30\!\cdots\!68}{22\!\cdots\!43}a^{29}-\frac{404800465622850}{1297652289253}a^{26}-\frac{26\!\cdots\!39}{22\!\cdots\!43}a^{25}+\frac{25\!\cdots\!22}{1297652289253}a^{22}+\frac{16\!\cdots\!68}{22\!\cdots\!43}a^{21}-\frac{70\!\cdots\!59}{1297652289253}a^{18}-\frac{45\!\cdots\!31}{22\!\cdots\!43}a^{17}+\frac{14\!\cdots\!12}{1297652289253}a^{14}+\frac{91\!\cdots\!91}{22\!\cdots\!43}a^{13}-\frac{65\!\cdots\!74}{1297652289253}a^{10}-\frac{42\!\cdots\!53}{22\!\cdots\!43}a^{9}+\frac{24\!\cdots\!88}{1297652289253}a^{6}+\frac{16\!\cdots\!78}{22\!\cdots\!43}a^{5}-\frac{277613222772}{1297652289253}a^{2}-\frac{10\!\cdots\!25}{22\!\cdots\!43}a-1$, $\frac{64\!\cdots\!51}{22\!\cdots\!43}a^{39}+\frac{54\!\cdots\!97}{22\!\cdots\!43}a^{38}-\frac{53\!\cdots\!40}{22\!\cdots\!43}a^{37}-\frac{16\!\cdots\!50}{22\!\cdots\!43}a^{35}-\frac{13\!\cdots\!60}{22\!\cdots\!43}a^{34}+\frac{13\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{28\!\cdots\!71}{22\!\cdots\!43}a^{31}+\frac{23\!\cdots\!42}{22\!\cdots\!43}a^{30}-\frac{23\!\cdots\!43}{22\!\cdots\!43}a^{29}-\frac{24\!\cdots\!75}{22\!\cdots\!43}a^{27}-\frac{19\!\cdots\!90}{22\!\cdots\!43}a^{26}+\frac{20\!\cdots\!20}{22\!\cdots\!43}a^{25}+\frac{15\!\cdots\!04}{22\!\cdots\!43}a^{23}+\frac{12\!\cdots\!42}{22\!\cdots\!43}a^{22}-\frac{12\!\cdots\!92}{22\!\cdots\!43}a^{21}-\frac{42\!\cdots\!34}{22\!\cdots\!43}a^{19}-\frac{31\!\cdots\!34}{22\!\cdots\!43}a^{18}+\frac{35\!\cdots\!64}{22\!\cdots\!43}a^{17}+\frac{86\!\cdots\!01}{22\!\cdots\!43}a^{15}+\frac{24\!\cdots\!27}{94\!\cdots\!23}a^{14}-\frac{71\!\cdots\!12}{22\!\cdots\!43}a^{13}-\frac{40\!\cdots\!30}{22\!\cdots\!43}a^{11}-\frac{59\!\cdots\!92}{22\!\cdots\!43}a^{10}+\frac{32\!\cdots\!84}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!28}{22\!\cdots\!43}a^{7}+\frac{38\!\cdots\!10}{22\!\cdots\!43}a^{6}-\frac{12\!\cdots\!81}{22\!\cdots\!43}a^{5}-\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}+\frac{47\!\cdots\!33}{22\!\cdots\!43}a^{2}+\frac{13\!\cdots\!52}{22\!\cdots\!43}a$, $\frac{65\!\cdots\!20}{22\!\cdots\!43}a^{39}-\frac{39\!\cdots\!66}{17\!\cdots\!71}a^{38}-\frac{12\!\cdots\!62}{22\!\cdots\!43}a^{37}-\frac{16\!\cdots\!10}{22\!\cdots\!43}a^{35}+\frac{98\!\cdots\!00}{17\!\cdots\!71}a^{34}+\frac{30\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{29\!\cdots\!33}{22\!\cdots\!43}a^{31}-\frac{17\!\cdots\!06}{17\!\cdots\!71}a^{30}-\frac{52\!\cdots\!45}{22\!\cdots\!43}a^{29}-\frac{24\!\cdots\!70}{22\!\cdots\!43}a^{27}+\frac{14\!\cdots\!09}{17\!\cdots\!71}a^{26}+\frac{44\!\cdots\!23}{22\!\cdots\!43}a^{25}+\frac{15\!\cdots\!66}{22\!\cdots\!43}a^{23}-\frac{92\!\cdots\!59}{17\!\cdots\!71}a^{22}-\frac{27\!\cdots\!45}{22\!\cdots\!43}a^{21}-\frac{43\!\cdots\!08}{22\!\cdots\!43}a^{19}+\frac{26\!\cdots\!03}{17\!\cdots\!71}a^{18}+\frac{70\!\cdots\!65}{22\!\cdots\!43}a^{17}+\frac{87\!\cdots\!16}{22\!\cdots\!43}a^{15}-\frac{52\!\cdots\!09}{17\!\cdots\!71}a^{14}-\frac{54\!\cdots\!15}{94\!\cdots\!23}a^{13}-\frac{40\!\cdots\!42}{22\!\cdots\!43}a^{11}+\frac{24\!\cdots\!58}{17\!\cdots\!71}a^{10}+\frac{13\!\cdots\!70}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!38}{22\!\cdots\!43}a^{7}-\frac{92\!\cdots\!82}{17\!\cdots\!71}a^{6}-\frac{86\!\cdots\!75}{22\!\cdots\!43}a^{5}-\frac{17\!\cdots\!76}{22\!\cdots\!43}a^{3}+\frac{60\!\cdots\!05}{17\!\cdots\!71}a^{2}-\frac{79\!\cdots\!13}{22\!\cdots\!43}a$, $\frac{64\!\cdots\!51}{22\!\cdots\!43}a^{39}+\frac{56\!\cdots\!22}{22\!\cdots\!43}a^{38}-\frac{53\!\cdots\!40}{22\!\cdots\!43}a^{37}-\frac{16\!\cdots\!50}{22\!\cdots\!43}a^{35}-\frac{13\!\cdots\!60}{22\!\cdots\!43}a^{34}+\frac{13\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{28\!\cdots\!71}{22\!\cdots\!43}a^{31}+\frac{24\!\cdots\!32}{22\!\cdots\!43}a^{30}-\frac{23\!\cdots\!43}{22\!\cdots\!43}a^{29}-\frac{24\!\cdots\!75}{22\!\cdots\!43}a^{27}-\frac{20\!\cdots\!28}{22\!\cdots\!43}a^{26}+\frac{20\!\cdots\!20}{22\!\cdots\!43}a^{25}+\frac{15\!\cdots\!04}{22\!\cdots\!43}a^{23}+\frac{12\!\cdots\!32}{22\!\cdots\!43}a^{22}-\frac{12\!\cdots\!92}{22\!\cdots\!43}a^{21}-\frac{42\!\cdots\!34}{22\!\cdots\!43}a^{19}-\frac{32\!\cdots\!64}{22\!\cdots\!43}a^{18}+\frac{35\!\cdots\!64}{22\!\cdots\!43}a^{17}+\frac{86\!\cdots\!01}{22\!\cdots\!43}a^{15}+\frac{25\!\cdots\!12}{94\!\cdots\!23}a^{14}-\frac{71\!\cdots\!12}{22\!\cdots\!43}a^{13}-\frac{40\!\cdots\!30}{22\!\cdots\!43}a^{11}-\frac{60\!\cdots\!32}{22\!\cdots\!43}a^{10}+\frac{32\!\cdots\!84}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!28}{22\!\cdots\!43}a^{7}+\frac{39\!\cdots\!60}{22\!\cdots\!43}a^{6}-\frac{12\!\cdots\!81}{22\!\cdots\!43}a^{5}-\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}+\frac{53\!\cdots\!83}{22\!\cdots\!43}a^{2}+\frac{13\!\cdots\!52}{22\!\cdots\!43}a$, $\frac{84\!\cdots\!40}{22\!\cdots\!43}a^{39}+\frac{106777091540}{1297652289253}a^{38}-\frac{166569250115625}{25\!\cdots\!51}a^{36}-\frac{21\!\cdots\!25}{22\!\cdots\!43}a^{35}-\frac{2668717678430}{1297652289253}a^{34}+\frac{41\!\cdots\!84}{25\!\cdots\!51}a^{32}+\frac{37\!\cdots\!80}{22\!\cdots\!43}a^{31}+\frac{47071002697755}{1297652289253}a^{30}-\frac{73\!\cdots\!45}{25\!\cdots\!51}a^{28}-\frac{32\!\cdots\!53}{22\!\cdots\!43}a^{27}-\frac{404800465622850}{1297652289253}a^{26}+\frac{63\!\cdots\!49}{25\!\cdots\!51}a^{24}+\frac{20\!\cdots\!41}{22\!\cdots\!43}a^{23}+\frac{25\!\cdots\!22}{1297652289253}a^{22}-\frac{39\!\cdots\!42}{25\!\cdots\!51}a^{20}-\frac{56\!\cdots\!13}{22\!\cdots\!43}a^{19}-\frac{70\!\cdots\!59}{1297652289253}a^{18}+\frac{11\!\cdots\!74}{25\!\cdots\!51}a^{16}+\frac{11\!\cdots\!21}{22\!\cdots\!43}a^{15}+\frac{14\!\cdots\!12}{1297652289253}a^{14}-\frac{22\!\cdots\!23}{25\!\cdots\!51}a^{12}-\frac{52\!\cdots\!47}{22\!\cdots\!43}a^{11}-\frac{65\!\cdots\!74}{1297652289253}a^{10}+\frac{10\!\cdots\!43}{25\!\cdots\!51}a^{8}+\frac{20\!\cdots\!98}{22\!\cdots\!43}a^{7}+\frac{24\!\cdots\!88}{1297652289253}a^{6}-\frac{39\!\cdots\!24}{25\!\cdots\!51}a^{4}-\frac{13\!\cdots\!15}{22\!\cdots\!43}a^{3}-\frac{277613222772}{1297652289253}a^{2}+\frac{432993263941969}{25\!\cdots\!51}$, $\frac{28\!\cdots\!61}{22\!\cdots\!43}a^{38}+\frac{14\!\cdots\!74}{22\!\cdots\!43}a^{37}-\frac{69\!\cdots\!80}{22\!\cdots\!43}a^{34}-\frac{35\!\cdots\!80}{22\!\cdots\!43}a^{33}+\frac{12\!\cdots\!66}{22\!\cdots\!43}a^{30}+\frac{62\!\cdots\!31}{22\!\cdots\!43}a^{29}-\frac{10\!\cdots\!14}{22\!\cdots\!43}a^{26}-\frac{52\!\cdots\!53}{22\!\cdots\!43}a^{25}+\frac{62\!\cdots\!66}{22\!\cdots\!43}a^{22}+\frac{31\!\cdots\!31}{22\!\cdots\!43}a^{21}-\frac{16\!\cdots\!82}{22\!\cdots\!43}a^{18}-\frac{82\!\cdots\!87}{22\!\cdots\!43}a^{17}+\frac{12\!\cdots\!06}{94\!\cdots\!23}a^{14}+\frac{64\!\cdots\!06}{94\!\cdots\!23}a^{13}-\frac{30\!\cdots\!16}{22\!\cdots\!43}a^{10}-\frac{15\!\cdots\!06}{22\!\cdots\!43}a^{9}+\frac{19\!\cdots\!30}{22\!\cdots\!43}a^{6}+\frac{10\!\cdots\!05}{22\!\cdots\!43}a^{5}+\frac{28\!\cdots\!63}{22\!\cdots\!43}a^{2}+\frac{13\!\cdots\!00}{22\!\cdots\!43}a+1$, $\frac{64\!\cdots\!51}{22\!\cdots\!43}a^{39}+\frac{35\!\cdots\!55}{22\!\cdots\!43}a^{38}+\frac{12\!\cdots\!62}{22\!\cdots\!43}a^{37}-\frac{16\!\cdots\!50}{22\!\cdots\!43}a^{35}-\frac{89\!\cdots\!48}{22\!\cdots\!43}a^{34}-\frac{30\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{28\!\cdots\!71}{22\!\cdots\!43}a^{31}+\frac{15\!\cdots\!25}{22\!\cdots\!43}a^{30}+\frac{52\!\cdots\!45}{22\!\cdots\!43}a^{29}-\frac{24\!\cdots\!75}{22\!\cdots\!43}a^{27}-\frac{13\!\cdots\!43}{22\!\cdots\!43}a^{26}-\frac{44\!\cdots\!23}{22\!\cdots\!43}a^{25}+\frac{15\!\cdots\!04}{22\!\cdots\!43}a^{23}+\frac{84\!\cdots\!58}{22\!\cdots\!43}a^{22}+\frac{27\!\cdots\!45}{22\!\cdots\!43}a^{21}-\frac{42\!\cdots\!34}{22\!\cdots\!43}a^{19}-\frac{23\!\cdots\!76}{22\!\cdots\!43}a^{18}-\frac{70\!\cdots\!65}{22\!\cdots\!43}a^{17}+\frac{86\!\cdots\!01}{22\!\cdots\!43}a^{15}+\frac{47\!\cdots\!05}{22\!\cdots\!43}a^{14}+\frac{54\!\cdots\!15}{94\!\cdots\!23}a^{13}-\frac{40\!\cdots\!30}{22\!\cdots\!43}a^{11}-\frac{21\!\cdots\!89}{22\!\cdots\!43}a^{10}-\frac{13\!\cdots\!70}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!28}{22\!\cdots\!43}a^{7}+\frac{83\!\cdots\!24}{22\!\cdots\!43}a^{6}+\frac{86\!\cdots\!75}{22\!\cdots\!43}a^{5}-\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}-\frac{93\!\cdots\!47}{22\!\cdots\!43}a^{2}+\frac{79\!\cdots\!13}{22\!\cdots\!43}a$, $\frac{64\!\cdots\!51}{22\!\cdots\!43}a^{39}-\frac{12\!\cdots\!62}{22\!\cdots\!43}a^{37}+\frac{29\!\cdots\!75}{22\!\cdots\!43}a^{36}-\frac{16\!\cdots\!50}{22\!\cdots\!43}a^{35}+\frac{30\!\cdots\!00}{22\!\cdots\!43}a^{33}-\frac{71\!\cdots\!80}{22\!\cdots\!43}a^{32}+\frac{28\!\cdots\!71}{22\!\cdots\!43}a^{31}-\frac{52\!\cdots\!45}{22\!\cdots\!43}a^{29}+\frac{12\!\cdots\!81}{22\!\cdots\!43}a^{28}-\frac{24\!\cdots\!75}{22\!\cdots\!43}a^{27}+\frac{44\!\cdots\!23}{22\!\cdots\!43}a^{25}-\frac{10\!\cdots\!78}{22\!\cdots\!43}a^{24}+\frac{15\!\cdots\!04}{22\!\cdots\!43}a^{23}-\frac{27\!\cdots\!45}{22\!\cdots\!43}a^{21}+\frac{64\!\cdots\!81}{22\!\cdots\!43}a^{20}-\frac{42\!\cdots\!34}{22\!\cdots\!43}a^{19}+\frac{70\!\cdots\!65}{22\!\cdots\!43}a^{17}-\frac{16\!\cdots\!37}{22\!\cdots\!43}a^{16}+\frac{86\!\cdots\!01}{22\!\cdots\!43}a^{15}-\frac{54\!\cdots\!15}{94\!\cdots\!23}a^{13}+\frac{13\!\cdots\!63}{94\!\cdots\!23}a^{12}-\frac{40\!\cdots\!30}{22\!\cdots\!43}a^{11}+\frac{13\!\cdots\!70}{22\!\cdots\!43}a^{9}-\frac{31\!\cdots\!06}{22\!\cdots\!43}a^{8}+\frac{15\!\cdots\!28}{22\!\cdots\!43}a^{7}-\frac{86\!\cdots\!75}{22\!\cdots\!43}a^{5}+\frac{20\!\cdots\!55}{22\!\cdots\!43}a^{4}-\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}-\frac{79\!\cdots\!13}{22\!\cdots\!43}a+\frac{64\!\cdots\!51}{22\!\cdots\!43}$, $\frac{18\!\cdots\!65}{22\!\cdots\!43}a^{39}-\frac{18\!\cdots\!79}{22\!\cdots\!43}a^{38}+\frac{166569250115625}{25\!\cdots\!51}a^{36}-\frac{46\!\cdots\!00}{22\!\cdots\!43}a^{35}+\frac{46\!\cdots\!50}{22\!\cdots\!43}a^{34}-\frac{41\!\cdots\!84}{25\!\cdots\!51}a^{32}+\frac{80\!\cdots\!25}{22\!\cdots\!43}a^{31}-\frac{81\!\cdots\!39}{22\!\cdots\!43}a^{30}+\frac{73\!\cdots\!45}{25\!\cdots\!51}a^{28}-\frac{68\!\cdots\!76}{22\!\cdots\!43}a^{27}+\frac{69\!\cdots\!36}{22\!\cdots\!43}a^{26}-\frac{63\!\cdots\!49}{25\!\cdots\!51}a^{24}+\frac{41\!\cdots\!25}{22\!\cdots\!43}a^{23}-\frac{43\!\cdots\!16}{22\!\cdots\!43}a^{22}+\frac{39\!\cdots\!42}{25\!\cdots\!51}a^{20}-\frac{10\!\cdots\!25}{22\!\cdots\!43}a^{19}+\frac{12\!\cdots\!47}{22\!\cdots\!43}a^{18}-\frac{11\!\cdots\!74}{25\!\cdots\!51}a^{16}+\frac{84\!\cdots\!33}{94\!\cdots\!23}a^{15}-\frac{24\!\cdots\!26}{22\!\cdots\!43}a^{14}+\frac{22\!\cdots\!23}{25\!\cdots\!51}a^{12}-\frac{20\!\cdots\!50}{22\!\cdots\!43}a^{11}+\frac{11\!\cdots\!78}{22\!\cdots\!43}a^{10}-\frac{10\!\cdots\!43}{25\!\cdots\!51}a^{8}+\frac{13\!\cdots\!75}{22\!\cdots\!43}a^{7}-\frac{43\!\cdots\!98}{22\!\cdots\!43}a^{6}+\frac{39\!\cdots\!24}{25\!\cdots\!51}a^{4}+\frac{16\!\cdots\!67}{22\!\cdots\!43}a^{3}+\frac{28\!\cdots\!95}{22\!\cdots\!43}a^{2}-\frac{25\!\cdots\!20}{25\!\cdots\!51}$, $\frac{65\!\cdots\!20}{22\!\cdots\!43}a^{39}+\frac{24\!\cdots\!25}{22\!\cdots\!43}a^{38}+\frac{53\!\cdots\!40}{22\!\cdots\!43}a^{37}-\frac{16\!\cdots\!10}{22\!\cdots\!43}a^{35}-\frac{61\!\cdots\!62}{22\!\cdots\!43}a^{34}-\frac{13\!\cdots\!00}{22\!\cdots\!43}a^{33}+\frac{29\!\cdots\!33}{22\!\cdots\!43}a^{31}+\frac{10\!\cdots\!65}{22\!\cdots\!43}a^{30}+\frac{23\!\cdots\!43}{22\!\cdots\!43}a^{29}-\frac{24\!\cdots\!70}{22\!\cdots\!43}a^{27}-\frac{93\!\cdots\!07}{22\!\cdots\!43}a^{26}-\frac{20\!\cdots\!20}{22\!\cdots\!43}a^{25}+\frac{15\!\cdots\!66}{22\!\cdots\!43}a^{23}+\frac{58\!\cdots\!16}{22\!\cdots\!43}a^{22}+\frac{12\!\cdots\!92}{22\!\cdots\!43}a^{21}-\frac{43\!\cdots\!08}{22\!\cdots\!43}a^{19}-\frac{16\!\cdots\!68}{22\!\cdots\!43}a^{18}-\frac{35\!\cdots\!64}{22\!\cdots\!43}a^{17}+\frac{87\!\cdots\!16}{22\!\cdots\!43}a^{15}+\frac{32\!\cdots\!99}{22\!\cdots\!43}a^{14}+\frac{71\!\cdots\!12}{22\!\cdots\!43}a^{13}-\frac{40\!\cdots\!42}{22\!\cdots\!43}a^{11}-\frac{15\!\cdots\!69}{22\!\cdots\!43}a^{10}-\frac{32\!\cdots\!84}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!38}{22\!\cdots\!43}a^{7}+\frac{56\!\cdots\!33}{22\!\cdots\!43}a^{6}+\frac{12\!\cdots\!81}{22\!\cdots\!43}a^{5}-\frac{17\!\cdots\!76}{22\!\cdots\!43}a^{3}-\frac{63\!\cdots\!77}{22\!\cdots\!43}a^{2}-\frac{13\!\cdots\!52}{22\!\cdots\!43}a$, $\frac{18\!\cdots\!79}{22\!\cdots\!43}a^{39}-\frac{106777091540}{1297652289253}a^{38}+\frac{166569250115625}{25\!\cdots\!51}a^{36}-\frac{46\!\cdots\!50}{22\!\cdots\!43}a^{35}+\frac{2668717678430}{1297652289253}a^{34}-\frac{41\!\cdots\!84}{25\!\cdots\!51}a^{32}+\frac{81\!\cdots\!39}{22\!\cdots\!43}a^{31}-\frac{47071002697755}{1297652289253}a^{30}+\frac{73\!\cdots\!45}{25\!\cdots\!51}a^{28}-\frac{69\!\cdots\!36}{22\!\cdots\!43}a^{27}+\frac{404800465622850}{1297652289253}a^{26}-\frac{63\!\cdots\!49}{25\!\cdots\!51}a^{24}+\frac{43\!\cdots\!16}{22\!\cdots\!43}a^{23}-\frac{25\!\cdots\!22}{1297652289253}a^{22}+\frac{39\!\cdots\!42}{25\!\cdots\!51}a^{20}-\frac{12\!\cdots\!47}{22\!\cdots\!43}a^{19}+\frac{70\!\cdots\!59}{1297652289253}a^{18}-\frac{11\!\cdots\!74}{25\!\cdots\!51}a^{16}+\frac{24\!\cdots\!26}{22\!\cdots\!43}a^{15}-\frac{14\!\cdots\!12}{1297652289253}a^{14}+\frac{22\!\cdots\!23}{25\!\cdots\!51}a^{12}-\frac{11\!\cdots\!78}{22\!\cdots\!43}a^{11}+\frac{65\!\cdots\!74}{1297652289253}a^{10}-\frac{10\!\cdots\!43}{25\!\cdots\!51}a^{8}+\frac{43\!\cdots\!98}{22\!\cdots\!43}a^{7}-\frac{24\!\cdots\!88}{1297652289253}a^{6}+\frac{39\!\cdots\!24}{25\!\cdots\!51}a^{4}-\frac{28\!\cdots\!95}{22\!\cdots\!43}a^{3}+\frac{277613222772}{1297652289253}a^{2}-\frac{432993263941969}{25\!\cdots\!51}$, $\frac{12\!\cdots\!00}{22\!\cdots\!43}a^{39}+\frac{106777091540}{1297652289253}a^{38}+\frac{166569250115625}{25\!\cdots\!51}a^{36}-\frac{31\!\cdots\!20}{22\!\cdots\!43}a^{35}-\frac{2668717678430}{1297652289253}a^{34}-\frac{41\!\cdots\!84}{25\!\cdots\!51}a^{32}+\frac{55\!\cdots\!23}{22\!\cdots\!43}a^{31}+\frac{47071002697755}{1297652289253}a^{30}+\frac{73\!\cdots\!45}{25\!\cdots\!51}a^{28}-\frac{47\!\cdots\!20}{22\!\cdots\!43}a^{27}-\frac{404800465622850}{1297652289253}a^{26}-\frac{63\!\cdots\!49}{25\!\cdots\!51}a^{24}+\frac{29\!\cdots\!40}{22\!\cdots\!43}a^{23}+\frac{25\!\cdots\!22}{1297652289253}a^{22}+\frac{39\!\cdots\!42}{25\!\cdots\!51}a^{20}-\frac{83\!\cdots\!52}{22\!\cdots\!43}a^{19}-\frac{70\!\cdots\!59}{1297652289253}a^{18}-\frac{11\!\cdots\!74}{25\!\cdots\!51}a^{16}+\frac{16\!\cdots\!20}{22\!\cdots\!43}a^{15}+\frac{14\!\cdots\!12}{1297652289253}a^{14}+\frac{22\!\cdots\!23}{25\!\cdots\!51}a^{12}-\frac{77\!\cdots\!00}{22\!\cdots\!43}a^{11}-\frac{65\!\cdots\!74}{1297652289253}a^{10}-\frac{10\!\cdots\!43}{25\!\cdots\!51}a^{8}+\frac{29\!\cdots\!95}{22\!\cdots\!43}a^{7}+\frac{24\!\cdots\!88}{1297652289253}a^{6}+\frac{39\!\cdots\!24}{25\!\cdots\!51}a^{4}-\frac{32\!\cdots\!00}{22\!\cdots\!43}a^{3}-\frac{277613222772}{1297652289253}a^{2}-\frac{25\!\cdots\!20}{25\!\cdots\!51}$ 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:  \( 7121371366090164.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 7121371366090164.0 \cdot 125}{24\cdot\sqrt{8900013646919506378267907122148784192909845412083315599549447274496}}\cr\approx \mathstrut & 0.114331114277240 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 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^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 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^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 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^40 - 25*x^36 + 441*x^32 - 3794*x^28 + 23626*x^24 - 66403*x^20 + 133864*x^16 - 62097*x^12 + 23622*x^8 - 155*x^4 + 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

$C_2^2\times C_{10}$ (as 40T7):

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)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{24})\), 10.0.219503494144.1, 10.0.1706859170463744.1, 10.10.1706859170463744.1, 10.0.7024111812608.1, 10.10.7024111812608.1, 10.10.53339349076992.1, 10.0.52089208083.1, 20.0.2983289065263288625233938941476864.4, 20.0.50522262278163705147147943936.1, 20.0.2845086159957207322343768064.1, 20.0.2983289065263288625233938941476864.1, 20.0.2913368227796180298080018497536.4, 20.0.2913368227796180298080018497536.2, 20.20.2983289065263288625233938941476864.1

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 R ${\href{/padicField/5.10.0.1}{10} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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
\(2\) Copy content Toggle raw display Deg $40$$4$$10$$80$
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) 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}$
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}$