Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 1)
gp: K = bnfinit(y^20 - 53*y^18 + 1034*y^16 - 9069*y^14 + 35792*y^12 - 63688*y^10 + 50170*y^8 - 18280*y^6 + 3064*y^4 - 197*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 1)
\( x^{20} - 53 x^{18} + 1034 x^{16} - 9069 x^{14} + 35792 x^{12} - 63688 x^{10} + 50170 x^{8} - 18280 x^{6} + \cdots + 1 \)
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: | $[20, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(48594796132978115111671719150390625\)
\(\medspace = 5^{10}\cdot 17^{8}\cdot 61^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.24\) | 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: | $5^{1/2}17^{1/2}61^{1/2}\approx 72.00694410957877$ | ||
| Ramified primes: |
\(5\), \(17\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{1190}a^{14}+\frac{47}{1190}a^{12}+\frac{191}{1190}a^{10}-\frac{1}{2}a^{9}+\frac{59}{1190}a^{8}-\frac{1}{2}a^{7}-\frac{4}{595}a^{6}-\frac{1}{2}a^{5}+\frac{92}{595}a^{4}-\frac{1}{2}a^{3}+\frac{15}{238}a^{2}+\frac{11}{1190}$, $\frac{1}{1190}a^{15}+\frac{47}{1190}a^{13}+\frac{191}{1190}a^{11}-\frac{268}{595}a^{9}-\frac{1}{2}a^{8}+\frac{587}{1190}a^{7}-\frac{1}{2}a^{6}+\frac{92}{595}a^{5}-\frac{1}{2}a^{4}+\frac{15}{238}a^{3}-\frac{292}{595}a-\frac{1}{2}$, $\frac{1}{1190}a^{16}-\frac{233}{1190}a^{12}+\frac{1}{170}a^{10}-\frac{401}{1190}a^{8}-\frac{1}{2}a^{7}+\frac{8}{17}a^{6}-\frac{243}{1190}a^{4}-\frac{77}{170}a^{2}-\frac{1}{2}a+\frac{39}{595}$, $\frac{1}{1190}a^{17}-\frac{233}{1190}a^{13}+\frac{1}{170}a^{11}-\frac{401}{1190}a^{9}-\frac{1}{2}a^{8}+\frac{8}{17}a^{7}-\frac{243}{1190}a^{5}-\frac{77}{170}a^{3}-\frac{1}{2}a^{2}+\frac{39}{595}a$, $\frac{1}{22956290}a^{18}+\frac{8207}{22956290}a^{16}+\frac{228}{2295629}a^{14}+\frac{1801566}{11478145}a^{12}-\frac{1918407}{11478145}a^{10}-\frac{1}{2}a^{9}+\frac{681873}{4591258}a^{8}-\frac{5711977}{22956290}a^{6}-\frac{1}{2}a^{5}+\frac{273918}{675185}a^{4}-\frac{1}{2}a^{3}-\frac{930918}{2295629}a^{2}-\frac{3836361}{22956290}$, $\frac{1}{22956290}a^{19}+\frac{8207}{22956290}a^{17}+\frac{228}{2295629}a^{15}+\frac{1801566}{11478145}a^{13}-\frac{1918407}{11478145}a^{11}-\frac{806878}{2295629}a^{9}+\frac{2883084}{11478145}a^{7}-\frac{1}{2}a^{6}+\frac{273918}{675185}a^{5}-\frac{1}{2}a^{4}-\frac{930918}{2295629}a^{3}+\frac{3820892}{11478145}a-\frac{1}{2}$
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: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{2741}{22729}a^{19}-\frac{1452373}{227290}a^{17}+\frac{2832299}{22729}a^{15}-\frac{35458663}{32470}a^{13}+\frac{488896297}{113645}a^{11}-\frac{866338696}{113645}a^{9}+\frac{270327153}{45458}a^{7}-\frac{68868013}{32470}a^{5}+\frac{10954621}{32470}a^{3}-\frac{1912482}{113645}a-\frac{1}{2}$, $a$, $\frac{18792989}{11478145}a^{19}-\frac{10562403}{22956290}a^{18}-\frac{58416088}{675185}a^{17}+\frac{279290632}{11478145}a^{16}+\frac{226772551}{135037}a^{15}-\frac{155093745}{327947}a^{14}-\frac{66958228801}{4591258}a^{13}+\frac{13503398077}{3279470}a^{12}+\frac{646179188836}{11478145}a^{11}-\frac{366953856153}{22956290}a^{10}-\frac{156300211478}{1639735}a^{9}+\frac{125809839733}{4591258}a^{8}+\frac{1532868656379}{22956290}a^{7}-\frac{64711767257}{3279470}a^{6}-\frac{86523626453}{4591258}a^{5}+\frac{9603595991}{1639735}a^{4}+\frac{39782536807}{22956290}a^{3}-\frac{2692234437}{4591258}a^{2}+\frac{52636509}{11478145}a+\frac{66753153}{22956290}$, $\frac{2579524}{11478145}a^{19}-\frac{10253436}{11478145}a^{18}-\frac{272489641}{22956290}a^{17}+\frac{541418239}{11478145}a^{16}+\frac{1056923037}{4591258}a^{15}-\frac{20991438073}{22956290}a^{14}-\frac{2694956059}{1350370}a^{13}+\frac{90927248107}{11478145}a^{12}+\frac{88052765572}{11478145}a^{11}-\frac{20537499049}{675185}a^{10}-\frac{147380365179}{11478145}a^{9}+\frac{1168983028791}{22956290}a^{8}+\frac{99300982307}{11478145}a^{7}-\frac{799535066907}{22956290}a^{6}-\frac{1398056332}{675185}a^{5}+\frac{109109030699}{11478145}a^{4}+\frac{10819777}{675185}a^{3}-\frac{10037637279}{11478145}a^{2}+\frac{51425699}{1639735}a+\frac{802177}{327947}$, $\frac{60636343}{22956290}a^{19}+\frac{90627}{675185}a^{18}-\frac{458014387}{3279470}a^{17}-\frac{162143019}{22956290}a^{16}+\frac{62295308153}{22956290}a^{15}+\frac{446359151}{3279470}a^{14}-\frac{542104370683}{22956290}a^{13}-\frac{5350477005}{4591258}a^{12}+\frac{2102654658291}{22956290}a^{11}+\frac{9998333778}{2295629}a^{10}-\frac{3601990128753}{22956290}a^{9}-\frac{78442016686}{11478145}a^{8}+\frac{260680186979}{2295629}a^{7}+\frac{6386955582}{1639735}a^{6}-\frac{810039147917}{22956290}a^{5}-\frac{12772058101}{22956290}a^{4}+\frac{20778559849}{4591258}a^{3}-\frac{207106671}{2295629}a^{2}-\frac{854515103}{4591258}a+\frac{209863167}{11478145}$, $\frac{9193799}{11478145}a^{19}-\frac{69364527}{1639735}a^{17}+\frac{9415677363}{11478145}a^{15}-\frac{16324627670}{2295629}a^{13}+\frac{44843854004}{1639735}a^{11}-\frac{31043970798}{675185}a^{9}+\frac{365910612108}{11478145}a^{7}-\frac{104606990434}{11478145}a^{5}+\frac{11170573828}{11478145}a^{3}-\frac{250025157}{11478145}a$, $\frac{4564369}{22956290}a^{19}+\frac{320756}{675185}a^{18}-\frac{120206358}{11478145}a^{17}-\frac{288048013}{11478145}a^{16}+\frac{2320321419}{11478145}a^{15}+\frac{1117572570}{2295629}a^{14}-\frac{39872014239}{22956290}a^{13}-\frac{1384951030}{327947}a^{12}+\frac{150305515109}{22956290}a^{11}+\frac{373319412411}{22956290}a^{10}-\frac{120793921877}{11478145}a^{9}-\frac{628714125491}{22956290}a^{8}+\frac{8851845679}{1350370}a^{7}+\frac{435187131667}{22956290}a^{6}-\frac{34915769433}{22956290}a^{5}-\frac{3445533099}{655894}a^{4}+\frac{2515106371}{22956290}a^{3}+\frac{11399315341}{22956290}a^{2}+\frac{49721367}{22956290}a-\frac{30936343}{11478145}$, $\frac{5298129}{11478145}a^{19}+\frac{166591}{3279470}a^{18}-\frac{3313916}{135037}a^{17}-\frac{30590997}{11478145}a^{16}+\frac{1104969046}{2295629}a^{15}+\frac{234580617}{4591258}a^{14}-\frac{48951900793}{11478145}a^{13}-\frac{4969596686}{11478145}a^{12}+\frac{197458125163}{11478145}a^{11}+\frac{36240233121}{22956290}a^{10}-\frac{735015780969}{22956290}a^{9}-\frac{53297094687}{22956290}a^{8}+\frac{632494078989}{22956290}a^{7}+\frac{23926432981}{22956290}a^{6}-\frac{131442164828}{11478145}a^{5}+\frac{1505806053}{22956290}a^{4}+\frac{1509312956}{675185}a^{3}-\frac{1145334299}{11478145}a^{2}-\frac{1875294308}{11478145}a+\frac{283752829}{22956290}$, $\frac{5298129}{11478145}a^{19}-\frac{166591}{3279470}a^{18}-\frac{3313916}{135037}a^{17}+\frac{30590997}{11478145}a^{16}+\frac{1104969046}{2295629}a^{15}-\frac{234580617}{4591258}a^{14}-\frac{48951900793}{11478145}a^{13}+\frac{4969596686}{11478145}a^{12}+\frac{197458125163}{11478145}a^{11}-\frac{36240233121}{22956290}a^{10}-\frac{735015780969}{22956290}a^{9}+\frac{53297094687}{22956290}a^{8}+\frac{632494078989}{22956290}a^{7}-\frac{23926432981}{22956290}a^{6}-\frac{131442164828}{11478145}a^{5}-\frac{1505806053}{22956290}a^{4}+\frac{1509312956}{675185}a^{3}+\frac{1145334299}{11478145}a^{2}-\frac{1875294308}{11478145}a-\frac{283752829}{22956290}$, $\frac{982490}{2295629}a^{19}+\frac{7348042}{11478145}a^{18}-\frac{517395813}{22956290}a^{17}-\frac{55442649}{1639735}a^{16}+\frac{4991731978}{11478145}a^{15}+\frac{7526689042}{11478145}a^{14}-\frac{85701195409}{22956290}a^{13}-\frac{130517204917}{22956290}a^{12}+\frac{1895338800}{135037}a^{11}+\frac{7172154796}{327947}a^{10}-\frac{256405910979}{11478145}a^{9}-\frac{168752366691}{4591258}a^{8}+\frac{152211596941}{11478145}a^{7}+\frac{58205231713}{2295629}a^{6}-\frac{51522728477}{22956290}a^{5}-\frac{80518983177}{11478145}a^{4}-\frac{2226970734}{11478145}a^{3}+\frac{7515699273}{11478145}a^{2}+\frac{590666961}{11478145}a-\frac{30876761}{11478145}$, $\frac{982490}{2295629}a^{19}-\frac{7348042}{11478145}a^{18}-\frac{517395813}{22956290}a^{17}+\frac{55442649}{1639735}a^{16}+\frac{4991731978}{11478145}a^{15}-\frac{7526689042}{11478145}a^{14}-\frac{85701195409}{22956290}a^{13}+\frac{130517204917}{22956290}a^{12}+\frac{1895338800}{135037}a^{11}-\frac{7172154796}{327947}a^{10}-\frac{256405910979}{11478145}a^{9}+\frac{168752366691}{4591258}a^{8}+\frac{152211596941}{11478145}a^{7}-\frac{58205231713}{2295629}a^{6}-\frac{51522728477}{22956290}a^{5}+\frac{80518983177}{11478145}a^{4}-\frac{2226970734}{11478145}a^{3}-\frac{7515699273}{11478145}a^{2}+\frac{590666961}{11478145}a+\frac{30876761}{11478145}$, $\frac{13687087}{22956290}a^{19}-\frac{2130705}{2295629}a^{18}-\frac{722088841}{22956290}a^{17}+\frac{224986021}{4591258}a^{16}+\frac{6988342982}{11478145}a^{15}-\frac{2180223690}{2295629}a^{14}-\frac{17245389829}{3279470}a^{13}+\frac{18879279341}{2295629}a^{12}+\frac{460288503361}{22956290}a^{11}-\frac{144841101953}{4591258}a^{10}-\frac{378792304637}{11478145}a^{9}+\frac{120996366749}{2295629}a^{8}+\frac{493901778479}{22956290}a^{7}-\frac{164898839467}{4591258}a^{6}-\frac{1651293083}{327947}a^{5}+\frac{44940047833}{4591258}a^{4}+\frac{767128489}{4591258}a^{3}-\frac{2120465721}{2295629}a^{2}+\frac{445701001}{11478145}a+\frac{36423115}{4591258}$, $\frac{8525498}{11478145}a^{19}+\frac{856456}{11478145}a^{18}-\frac{449888981}{11478145}a^{17}-\frac{45422546}{11478145}a^{16}+\frac{1244542756}{1639735}a^{15}+\frac{887123422}{11478145}a^{14}-\frac{75311954664}{11478145}a^{13}-\frac{445448145}{655894}a^{12}+\frac{575599897657}{22956290}a^{11}+\frac{30866815452}{11478145}a^{10}-\frac{190659711785}{4591258}a^{9}-\frac{3240853542}{675185}a^{8}+\frac{45391382009}{1639735}a^{7}+\frac{42895057587}{11478145}a^{6}-\frac{16460107749}{2295629}a^{5}-\frac{2010799888}{1639735}a^{4}+\frac{13717623411}{22956290}a^{3}+\frac{3032535479}{22956290}a^{2}+\frac{27808738}{11478145}a-\frac{667589}{675185}$, $\frac{12441197}{22956290}a^{19}-\frac{6070714}{11478145}a^{18}-\frac{657288671}{22956290}a^{17}+\frac{641203639}{22956290}a^{16}+\frac{12753549241}{22956290}a^{15}-\frac{365682071}{675185}a^{14}-\frac{15811857837}{3279470}a^{13}+\frac{107763524571}{22956290}a^{12}+\frac{213340860024}{11478145}a^{11}-\frac{207112125296}{11478145}a^{10}-\frac{360355134493}{11478145}a^{9}+\frac{695088113689}{22956290}a^{8}+\frac{251801014609}{11478145}a^{7}-\frac{239008022886}{11478145}a^{6}-\frac{10286075443}{1639735}a^{5}+\frac{131887874003}{22956290}a^{4}+\frac{2986975279}{4591258}a^{3}-\frac{1244732670}{2295629}a^{2}-\frac{140437123}{11478145}a+\frac{48613742}{11478145}$, $\frac{2029487}{3279470}a^{19}-\frac{6650657}{22956290}a^{18}-\frac{150290093}{4591258}a^{17}+\frac{175622319}{11478145}a^{16}+\frac{859450427}{1350370}a^{15}-\frac{6811285891}{22956290}a^{14}-\frac{25461298689}{4591258}a^{13}+\frac{59044939343}{22956290}a^{12}+\frac{495134691237}{22956290}a^{11}-\frac{113517498412}{11478145}a^{10}-\frac{426462900928}{11478145}a^{9}+\frac{381271415429}{22956290}a^{8}+\frac{62366312383}{2295629}a^{7}-\frac{262685878073}{22956290}a^{6}-\frac{5746199758}{675185}a^{5}+\frac{4274060423}{1350370}a^{4}+\frac{12225250931}{11478145}a^{3}-\frac{3358037754}{11478145}a^{2}-\frac{849297199}{22956290}a+\frac{555697}{22956290}$, $\frac{33351047}{22956290}a^{19}+\frac{568993}{675185}a^{18}-\frac{351931247}{4591258}a^{17}-\frac{510965709}{11478145}a^{16}+\frac{28626621}{19291}a^{15}+\frac{9912104624}{11478145}a^{14}-\frac{294346943029}{22956290}a^{13}-\frac{171967554809}{22956290}a^{12}+\frac{561827787257}{11478145}a^{11}+\frac{662229707361}{22956290}a^{10}-\frac{132656581309}{1639735}a^{9}-\frac{557851917523}{11478145}a^{8}+\frac{88145924094}{1639735}a^{7}+\frac{774048151327}{22956290}a^{6}-\frac{160364252437}{11478145}a^{5}-\frac{108281626694}{11478145}a^{4}+\frac{28465316531}{22956290}a^{3}+\frac{1465138652}{1639735}a^{2}-\frac{124874487}{11478145}a-\frac{3471227}{655894}$, $\frac{16516558}{11478145}a^{19}-\frac{8394559}{11478145}a^{18}-\frac{102609183}{1350370}a^{17}+\frac{886894719}{22956290}a^{16}+\frac{142094454}{96455}a^{15}-\frac{1228942791}{1639735}a^{14}-\frac{4186198807}{327947}a^{13}+\frac{74628460363}{11478145}a^{12}+\frac{1125705643287}{22956290}a^{11}-\frac{574833853677}{22956290}a^{10}-\frac{943514723082}{11478145}a^{9}+\frac{484299334212}{11478145}a^{8}+\frac{92610145543}{1639735}a^{7}-\frac{47994454198}{1639735}a^{6}-\frac{51480555353}{3279470}a^{5}+\frac{93814625994}{11478145}a^{4}+\frac{18056619736}{11478145}a^{3}-\frac{104476385}{135037}a^{2}-\frac{304561119}{11478145}a+\frac{121482649}{22956290}$, $\frac{33351047}{22956290}a^{19}-\frac{568993}{675185}a^{18}-\frac{351931247}{4591258}a^{17}+\frac{510965709}{11478145}a^{16}+\frac{28626621}{19291}a^{15}-\frac{9912104624}{11478145}a^{14}-\frac{294346943029}{22956290}a^{13}+\frac{171967554809}{22956290}a^{12}+\frac{561827787257}{11478145}a^{11}-\frac{662229707361}{22956290}a^{10}-\frac{132656581309}{1639735}a^{9}+\frac{557851917523}{11478145}a^{8}+\frac{88145924094}{1639735}a^{7}-\frac{774048151327}{22956290}a^{6}-\frac{160364252437}{11478145}a^{5}+\frac{108281626694}{11478145}a^{4}+\frac{28465316531}{22956290}a^{3}-\frac{1465138652}{1639735}a^{2}-\frac{124874487}{11478145}a+\frac{3471227}{655894}$, $\frac{31809543}{11478145}a^{19}-\frac{28793447}{22956290}a^{18}-\frac{3359426099}{22956290}a^{17}+\frac{760106483}{11478145}a^{16}+\frac{13025645579}{4591258}a^{15}-\frac{29464100947}{22956290}a^{14}-\frac{8061258419}{327947}a^{13}+\frac{255151957227}{22956290}a^{12}+\frac{2167311116359}{22956290}a^{11}-\frac{978826081999}{22956290}a^{10}-\frac{3630810644799}{22956290}a^{9}+\frac{1635280753277}{22956290}a^{8}+\frac{1244293393639}{11478145}a^{7}-\frac{111296123794}{2295629}a^{6}-\frac{19552803523}{655894}a^{5}+\frac{150397460504}{11478145}a^{4}+\frac{65298927999}{22956290}a^{3}-\frac{391336885}{327947}a^{2}-\frac{323082202}{11478145}a+\frac{14874850}{2295629}$
| 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: | \( 123847275257 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 123847275257 \cdot 1}{2\cdot\sqrt{48594796132978115111671719150390625}}\cr\approx \mathstrut & 0.294551671482931 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 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 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 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 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 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 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 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
$C_2\times D_{10}$ (as 20T8):
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 40 |
| The 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 5.5.26884225.1, 10.10.220442273924440625.1, 10.10.3613807769253125.1, 10.10.44088454784888125.1 |
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
| Galois closure: | deg 40 |
| Degree 20 siblings: | 20.20.3774226305410017540250773134765625.1, deg 20, deg 20 |
| Minimal sibling: | 20.20.3774226305410017540250773134765625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{10}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |