Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1)
gp: K = bnfinit(y^25 - 4*y^24 + 4*y^23 + 5*y^22 - 20*y^21 + 18*y^20 + 19*y^19 - 28*y^18 - 2*y^16 + 2*y^15 - 28*y^14 - 29*y^13 + 120*y^12 + 59*y^11 - 112*y^10 - 35*y^9 + 16*y^8 - 23*y^7 + 29*y^6 + 32*y^5 - 21*y^4 - 12*y^3 + 7*y^2 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1)
\( x^{25} - 4 x^{24} + 4 x^{23} + 5 x^{22} - 20 x^{21} + 18 x^{20} + 19 x^{19} - 28 x^{18} - 2 x^{16} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[5, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(190035222333323626806494766049\)
\(\medspace = 7^{10}\cdot 11^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.83\) | 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: | $7^{1/2}11^{4/5}\approx 18.016198912314337$ | ||
| Ramified primes: |
\(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $5$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{14\!\cdots\!82}a^{24}+\frac{140614422730571}{14\!\cdots\!82}a^{23}-\frac{104578792346784}{711214892241391}a^{22}-\frac{165233546399600}{711214892241391}a^{21}-\frac{144093055169530}{711214892241391}a^{20}-\frac{144067137096092}{711214892241391}a^{19}+\frac{328858474526599}{711214892241391}a^{18}+\frac{11388431288854}{711214892241391}a^{17}+\frac{554089679032639}{14\!\cdots\!82}a^{16}-\frac{249580344224563}{711214892241391}a^{15}-\frac{61285048475461}{711214892241391}a^{14}-\frac{195315440452465}{711214892241391}a^{13}+\frac{690630018986355}{14\!\cdots\!82}a^{12}+\frac{124898007688931}{711214892241391}a^{11}-\frac{428902554332149}{14\!\cdots\!82}a^{10}+\frac{638431435676121}{14\!\cdots\!82}a^{9}+\frac{261915975651208}{711214892241391}a^{8}+\frac{169378685340181}{14\!\cdots\!82}a^{7}+\frac{47853678613913}{14\!\cdots\!82}a^{6}-\frac{256967440247395}{711214892241391}a^{5}-\frac{168316319972885}{14\!\cdots\!82}a^{4}-\frac{249305403652881}{711214892241391}a^{3}+\frac{87804121646247}{711214892241391}a^{2}-\frac{188177807611685}{711214892241391}a+\frac{685877223418807}{14\!\cdots\!82}$
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: | $14$ | 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: |
$\frac{616740030838839}{14\!\cdots\!82}a^{24}-\frac{21\!\cdots\!53}{14\!\cdots\!82}a^{23}+\frac{650313690763444}{711214892241391}a^{22}+\frac{20\!\cdots\!93}{711214892241391}a^{21}-\frac{57\!\cdots\!49}{711214892241391}a^{20}+\frac{35\!\cdots\!75}{711214892241391}a^{19}+\frac{76\!\cdots\!89}{711214892241391}a^{18}-\frac{69\!\cdots\!45}{711214892241391}a^{17}+\frac{427851819936435}{14\!\cdots\!82}a^{16}-\frac{20\!\cdots\!49}{711214892241391}a^{15}-\frac{20\!\cdots\!94}{711214892241391}a^{14}-\frac{73\!\cdots\!65}{711214892241391}a^{13}-\frac{30\!\cdots\!75}{14\!\cdots\!82}a^{12}+\frac{31\!\cdots\!75}{711214892241391}a^{11}+\frac{60\!\cdots\!41}{14\!\cdots\!82}a^{10}-\frac{39\!\cdots\!09}{14\!\cdots\!82}a^{9}-\frac{93\!\cdots\!26}{711214892241391}a^{8}-\frac{88\!\cdots\!59}{14\!\cdots\!82}a^{7}-\frac{30\!\cdots\!67}{14\!\cdots\!82}a^{6}+\frac{62\!\cdots\!96}{711214892241391}a^{5}+\frac{18\!\cdots\!77}{14\!\cdots\!82}a^{4}-\frac{24\!\cdots\!09}{711214892241391}a^{3}-\frac{891669873380834}{711214892241391}a^{2}+\frac{319425313305334}{711214892241391}a-\frac{10\!\cdots\!61}{14\!\cdots\!82}$, $\frac{24016287797655}{14\!\cdots\!82}a^{24}-\frac{414891369178632}{711214892241391}a^{23}+\frac{17\!\cdots\!78}{711214892241391}a^{22}-\frac{25\!\cdots\!91}{711214892241391}a^{21}-\frac{327522364816845}{711214892241391}a^{20}+\frac{74\!\cdots\!10}{711214892241391}a^{19}-\frac{10\!\cdots\!21}{711214892241391}a^{18}+\frac{162664859143296}{711214892241391}a^{17}+\frac{17\!\cdots\!53}{14\!\cdots\!82}a^{16}-\frac{89\!\cdots\!65}{14\!\cdots\!82}a^{15}+\frac{97\!\cdots\!65}{14\!\cdots\!82}a^{14}-\frac{10\!\cdots\!47}{14\!\cdots\!82}a^{13}+\frac{12\!\cdots\!48}{711214892241391}a^{12}+\frac{18\!\cdots\!23}{711214892241391}a^{11}-\frac{89\!\cdots\!73}{14\!\cdots\!82}a^{10}+\frac{33\!\cdots\!37}{711214892241391}a^{9}+\frac{36\!\cdots\!80}{711214892241391}a^{8}-\frac{27\!\cdots\!01}{14\!\cdots\!82}a^{7}+\frac{23\!\cdots\!90}{711214892241391}a^{6}+\frac{45\!\cdots\!19}{711214892241391}a^{5}-\frac{32\!\cdots\!51}{14\!\cdots\!82}a^{4}-\frac{88\!\cdots\!95}{14\!\cdots\!82}a^{3}+\frac{12\!\cdots\!43}{14\!\cdots\!82}a^{2}+\frac{884414108898621}{14\!\cdots\!82}a-\frac{886644642906804}{711214892241391}$, $a$, $\frac{179926376817890}{711214892241391}a^{24}-\frac{884923375894511}{14\!\cdots\!82}a^{23}-\frac{520974428124481}{711214892241391}a^{22}+\frac{28\!\cdots\!01}{711214892241391}a^{21}-\frac{86\!\cdots\!09}{14\!\cdots\!82}a^{20}-\frac{355609003056282}{711214892241391}a^{19}+\frac{20\!\cdots\!67}{14\!\cdots\!82}a^{18}-\frac{14\!\cdots\!61}{14\!\cdots\!82}a^{17}-\frac{10\!\cdots\!75}{14\!\cdots\!82}a^{16}+\frac{455499156972919}{14\!\cdots\!82}a^{15}-\frac{71\!\cdots\!35}{14\!\cdots\!82}a^{14}-\frac{10\!\cdots\!06}{711214892241391}a^{13}-\frac{18\!\cdots\!37}{711214892241391}a^{12}+\frac{22\!\cdots\!07}{711214892241391}a^{11}+\frac{64\!\cdots\!91}{14\!\cdots\!82}a^{10}-\frac{36\!\cdots\!03}{14\!\cdots\!82}a^{9}-\frac{26\!\cdots\!09}{14\!\cdots\!82}a^{8}+\frac{53\!\cdots\!29}{14\!\cdots\!82}a^{7}-\frac{21\!\cdots\!25}{14\!\cdots\!82}a^{6}+\frac{29\!\cdots\!64}{711214892241391}a^{5}+\frac{21\!\cdots\!41}{14\!\cdots\!82}a^{4}-\frac{35\!\cdots\!00}{711214892241391}a^{3}-\frac{38\!\cdots\!81}{711214892241391}a^{2}+\frac{291768914220705}{14\!\cdots\!82}a+\frac{5439624931054}{711214892241391}$, $\frac{330296044591027}{711214892241391}a^{24}-\frac{15\!\cdots\!40}{711214892241391}a^{23}+\frac{47\!\cdots\!09}{14\!\cdots\!82}a^{22}+\frac{964695916961547}{14\!\cdots\!82}a^{21}-\frac{72\!\cdots\!42}{711214892241391}a^{20}+\frac{10\!\cdots\!26}{711214892241391}a^{19}+\frac{29\!\cdots\!27}{14\!\cdots\!82}a^{18}-\frac{11\!\cdots\!55}{711214892241391}a^{17}+\frac{59\!\cdots\!81}{14\!\cdots\!82}a^{16}-\frac{29\!\cdots\!19}{14\!\cdots\!82}a^{15}+\frac{36\!\cdots\!37}{711214892241391}a^{14}-\frac{10\!\cdots\!46}{711214892241391}a^{13}+\frac{10\!\cdots\!69}{711214892241391}a^{12}+\frac{93\!\cdots\!15}{14\!\cdots\!82}a^{11}-\frac{36\!\cdots\!63}{711214892241391}a^{10}-\frac{50\!\cdots\!95}{711214892241391}a^{9}-\frac{27\!\cdots\!30}{711214892241391}a^{8}+\frac{14\!\cdots\!95}{14\!\cdots\!82}a^{7}-\frac{77\!\cdots\!47}{14\!\cdots\!82}a^{6}+\frac{17\!\cdots\!98}{711214892241391}a^{5}+\frac{17\!\cdots\!77}{14\!\cdots\!82}a^{4}-\frac{10\!\cdots\!38}{711214892241391}a^{3}-\frac{21\!\cdots\!15}{711214892241391}a^{2}+\frac{53\!\cdots\!23}{14\!\cdots\!82}a-\frac{62344408233701}{14\!\cdots\!82}$, $\frac{533778876546055}{711214892241391}a^{24}-\frac{16\!\cdots\!05}{711214892241391}a^{23}+\frac{572161713199815}{14\!\cdots\!82}a^{22}+\frac{45\!\cdots\!56}{711214892241391}a^{21}-\frac{93\!\cdots\!98}{711214892241391}a^{20}+\frac{51\!\cdots\!27}{14\!\cdots\!82}a^{19}+\frac{16\!\cdots\!64}{711214892241391}a^{18}-\frac{19\!\cdots\!99}{14\!\cdots\!82}a^{17}-\frac{58\!\cdots\!71}{14\!\cdots\!82}a^{16}-\frac{91\!\cdots\!25}{14\!\cdots\!82}a^{15}-\frac{78\!\cdots\!19}{14\!\cdots\!82}a^{14}-\frac{23\!\cdots\!17}{14\!\cdots\!82}a^{13}-\frac{34\!\cdots\!09}{711214892241391}a^{12}+\frac{51\!\cdots\!08}{711214892241391}a^{11}+\frac{71\!\cdots\!19}{711214892241391}a^{10}-\frac{43\!\cdots\!29}{14\!\cdots\!82}a^{9}-\frac{64\!\cdots\!99}{14\!\cdots\!82}a^{8}-\frac{32\!\cdots\!17}{14\!\cdots\!82}a^{7}-\frac{46\!\cdots\!79}{14\!\cdots\!82}a^{6}+\frac{92\!\cdots\!49}{14\!\cdots\!82}a^{5}+\frac{21\!\cdots\!93}{711214892241391}a^{4}+\frac{50\!\cdots\!19}{14\!\cdots\!82}a^{3}-\frac{59\!\cdots\!29}{711214892241391}a^{2}-\frac{748300527756844}{711214892241391}a+\frac{21\!\cdots\!91}{14\!\cdots\!82}$, $\frac{651450101787365}{711214892241391}a^{24}-\frac{57\!\cdots\!49}{14\!\cdots\!82}a^{23}+\frac{74\!\cdots\!99}{14\!\cdots\!82}a^{22}+\frac{38\!\cdots\!77}{14\!\cdots\!82}a^{21}-\frac{28\!\cdots\!95}{14\!\cdots\!82}a^{20}+\frac{17\!\cdots\!29}{711214892241391}a^{19}+\frac{47\!\cdots\!36}{711214892241391}a^{18}-\frac{41\!\cdots\!35}{14\!\cdots\!82}a^{17}+\frac{10\!\cdots\!75}{711214892241391}a^{16}-\frac{88\!\cdots\!35}{711214892241391}a^{15}+\frac{10\!\cdots\!21}{14\!\cdots\!82}a^{14}-\frac{19\!\cdots\!13}{711214892241391}a^{13}-\frac{12\!\cdots\!85}{711214892241391}a^{12}+\frac{17\!\cdots\!59}{14\!\cdots\!82}a^{11}+\frac{56\!\cdots\!93}{14\!\cdots\!82}a^{10}-\frac{13\!\cdots\!59}{14\!\cdots\!82}a^{9}+\frac{10\!\cdots\!41}{14\!\cdots\!82}a^{8}-\frac{32\!\cdots\!09}{711214892241391}a^{7}-\frac{15\!\cdots\!19}{711214892241391}a^{6}+\frac{28\!\cdots\!53}{711214892241391}a^{5}+\frac{83\!\cdots\!60}{711214892241391}a^{4}-\frac{12\!\cdots\!21}{711214892241391}a^{3}+\frac{871027594974888}{711214892241391}a^{2}+\frac{23\!\cdots\!16}{711214892241391}a-\frac{494043898331477}{14\!\cdots\!82}$, $\frac{220626448628493}{14\!\cdots\!82}a^{24}-\frac{6478098949395}{14\!\cdots\!82}a^{23}-\frac{27\!\cdots\!11}{14\!\cdots\!82}a^{22}+\frac{57\!\cdots\!99}{14\!\cdots\!82}a^{21}-\frac{14\!\cdots\!02}{711214892241391}a^{20}-\frac{52\!\cdots\!54}{711214892241391}a^{19}+\frac{21\!\cdots\!03}{14\!\cdots\!82}a^{18}+\frac{239144037532436}{711214892241391}a^{17}-\frac{69\!\cdots\!12}{711214892241391}a^{16}+\frac{213766253771419}{14\!\cdots\!82}a^{15}-\frac{33\!\cdots\!77}{711214892241391}a^{14}-\frac{877987148434414}{711214892241391}a^{13}-\frac{36\!\cdots\!65}{14\!\cdots\!82}a^{12}+\frac{11\!\cdots\!05}{14\!\cdots\!82}a^{11}+\frac{10\!\cdots\!81}{14\!\cdots\!82}a^{10}+\frac{13\!\cdots\!35}{14\!\cdots\!82}a^{9}-\frac{33\!\cdots\!64}{711214892241391}a^{8}-\frac{68\!\cdots\!22}{711214892241391}a^{7}-\frac{77\!\cdots\!23}{711214892241391}a^{6}-\frac{72\!\cdots\!20}{711214892241391}a^{5}+\frac{13\!\cdots\!16}{711214892241391}a^{4}+\frac{56\!\cdots\!95}{711214892241391}a^{3}-\frac{53\!\cdots\!64}{711214892241391}a^{2}-\frac{16\!\cdots\!37}{14\!\cdots\!82}a+\frac{12\!\cdots\!20}{711214892241391}$, $\frac{445028178324855}{14\!\cdots\!82}a^{24}-\frac{420629018066846}{711214892241391}a^{23}-\frac{11\!\cdots\!37}{711214892241391}a^{22}+\frac{74\!\cdots\!79}{14\!\cdots\!82}a^{21}-\frac{58\!\cdots\!21}{14\!\cdots\!82}a^{20}-\frac{12\!\cdots\!09}{14\!\cdots\!82}a^{19}+\frac{16\!\cdots\!73}{711214892241391}a^{18}-\frac{22\!\cdots\!32}{711214892241391}a^{17}-\frac{14\!\cdots\!28}{711214892241391}a^{16}+\frac{99\!\cdots\!53}{14\!\cdots\!82}a^{15}-\frac{50\!\cdots\!67}{711214892241391}a^{14}-\frac{37\!\cdots\!25}{14\!\cdots\!82}a^{13}-\frac{41\!\cdots\!95}{14\!\cdots\!82}a^{12}+\frac{38\!\cdots\!45}{14\!\cdots\!82}a^{11}+\frac{76\!\cdots\!75}{711214892241391}a^{10}-\frac{37\!\cdots\!31}{14\!\cdots\!82}a^{9}-\frac{61\!\cdots\!61}{711214892241391}a^{8}+\frac{159246796994762}{711214892241391}a^{7}-\frac{61\!\cdots\!31}{711214892241391}a^{6}-\frac{58\!\cdots\!59}{14\!\cdots\!82}a^{5}+\frac{55\!\cdots\!21}{14\!\cdots\!82}a^{4}+\frac{90\!\cdots\!85}{14\!\cdots\!82}a^{3}-\frac{12\!\cdots\!51}{711214892241391}a^{2}-\frac{169206089843863}{711214892241391}a+\frac{41\!\cdots\!57}{14\!\cdots\!82}$, $\frac{435724237792987}{14\!\cdots\!82}a^{24}-\frac{555977673416710}{711214892241391}a^{23}-\frac{262483200495911}{711214892241391}a^{22}+\frac{38\!\cdots\!01}{14\!\cdots\!82}a^{21}-\frac{24\!\cdots\!73}{711214892241391}a^{20}-\frac{43\!\cdots\!75}{14\!\cdots\!82}a^{19}+\frac{17\!\cdots\!61}{14\!\cdots\!82}a^{18}+\frac{68451846960933}{14\!\cdots\!82}a^{17}-\frac{14\!\cdots\!99}{14\!\cdots\!82}a^{16}+\frac{39\!\cdots\!07}{14\!\cdots\!82}a^{15}-\frac{37\!\cdots\!53}{711214892241391}a^{14}-\frac{63\!\cdots\!85}{711214892241391}a^{13}-\frac{14\!\cdots\!98}{711214892241391}a^{12}+\frac{21\!\cdots\!89}{14\!\cdots\!82}a^{11}+\frac{94\!\cdots\!89}{14\!\cdots\!82}a^{10}-\frac{36\!\cdots\!49}{14\!\cdots\!82}a^{9}-\frac{51\!\cdots\!99}{14\!\cdots\!82}a^{8}-\frac{29\!\cdots\!23}{14\!\cdots\!82}a^{7}-\frac{12\!\cdots\!29}{711214892241391}a^{6}-\frac{10\!\cdots\!85}{14\!\cdots\!82}a^{5}+\frac{13\!\cdots\!90}{711214892241391}a^{4}+\frac{18\!\cdots\!80}{711214892241391}a^{3}-\frac{88\!\cdots\!99}{14\!\cdots\!82}a^{2}+\frac{720894959744548}{711214892241391}a+\frac{536578496868159}{711214892241391}$, $\frac{19\!\cdots\!95}{14\!\cdots\!82}a^{24}-\frac{65\!\cdots\!93}{14\!\cdots\!82}a^{23}+\frac{32\!\cdots\!29}{14\!\cdots\!82}a^{22}+\frac{14\!\cdots\!39}{14\!\cdots\!82}a^{21}-\frac{34\!\cdots\!87}{14\!\cdots\!82}a^{20}+\frac{78\!\cdots\!51}{711214892241391}a^{19}+\frac{27\!\cdots\!93}{711214892241391}a^{18}-\frac{38\!\cdots\!15}{14\!\cdots\!82}a^{17}-\frac{18\!\cdots\!43}{14\!\cdots\!82}a^{16}-\frac{50\!\cdots\!81}{14\!\cdots\!82}a^{15}-\frac{34\!\cdots\!41}{711214892241391}a^{14}-\frac{49\!\cdots\!13}{14\!\cdots\!82}a^{13}-\frac{46\!\cdots\!60}{711214892241391}a^{12}+\frac{19\!\cdots\!19}{14\!\cdots\!82}a^{11}+\frac{11\!\cdots\!42}{711214892241391}a^{10}-\frac{14\!\cdots\!17}{14\!\cdots\!82}a^{9}-\frac{13\!\cdots\!35}{14\!\cdots\!82}a^{8}-\frac{96\!\cdots\!25}{14\!\cdots\!82}a^{7}-\frac{32\!\cdots\!83}{711214892241391}a^{6}+\frac{15\!\cdots\!03}{711214892241391}a^{5}+\frac{85\!\cdots\!65}{14\!\cdots\!82}a^{4}-\frac{11\!\cdots\!53}{14\!\cdots\!82}a^{3}-\frac{28\!\cdots\!81}{14\!\cdots\!82}a^{2}+\frac{40\!\cdots\!09}{14\!\cdots\!82}a+\frac{37\!\cdots\!87}{14\!\cdots\!82}$, $\frac{14\!\cdots\!53}{14\!\cdots\!82}a^{24}-\frac{69\!\cdots\!43}{14\!\cdots\!82}a^{23}+\frac{57\!\cdots\!69}{711214892241391}a^{22}-\frac{10\!\cdots\!43}{711214892241391}a^{21}-\frac{27\!\cdots\!67}{14\!\cdots\!82}a^{20}+\frac{24\!\cdots\!76}{711214892241391}a^{19}-\frac{12\!\cdots\!75}{14\!\cdots\!82}a^{18}-\frac{31\!\cdots\!89}{14\!\cdots\!82}a^{17}+\frac{14\!\cdots\!33}{711214892241391}a^{16}-\frac{12\!\cdots\!41}{711214892241391}a^{15}+\frac{10\!\cdots\!33}{711214892241391}a^{14}-\frac{56\!\cdots\!55}{14\!\cdots\!82}a^{13}+\frac{10\!\cdots\!25}{711214892241391}a^{12}+\frac{85\!\cdots\!48}{711214892241391}a^{11}-\frac{29\!\cdots\!06}{711214892241391}a^{10}-\frac{11\!\cdots\!19}{14\!\cdots\!82}a^{9}+\frac{59\!\cdots\!95}{14\!\cdots\!82}a^{8}-\frac{10\!\cdots\!46}{711214892241391}a^{7}-\frac{21\!\cdots\!29}{14\!\cdots\!82}a^{6}+\frac{30\!\cdots\!46}{711214892241391}a^{5}-\frac{52\!\cdots\!92}{711214892241391}a^{4}-\frac{26\!\cdots\!41}{14\!\cdots\!82}a^{3}+\frac{95\!\cdots\!81}{14\!\cdots\!82}a^{2}+\frac{25\!\cdots\!80}{711214892241391}a-\frac{487732329563466}{711214892241391}$, $\frac{366333060423747}{14\!\cdots\!82}a^{24}-\frac{13\!\cdots\!60}{711214892241391}a^{23}+\frac{60\!\cdots\!27}{14\!\cdots\!82}a^{22}-\frac{13\!\cdots\!77}{711214892241391}a^{21}-\frac{63\!\cdots\!87}{711214892241391}a^{20}+\frac{28\!\cdots\!29}{14\!\cdots\!82}a^{19}-\frac{66\!\cdots\!16}{711214892241391}a^{18}-\frac{30\!\cdots\!47}{14\!\cdots\!82}a^{17}+\frac{14\!\cdots\!97}{711214892241391}a^{16}-\frac{13\!\cdots\!52}{711214892241391}a^{15}+\frac{32\!\cdots\!29}{711214892241391}a^{14}-\frac{46\!\cdots\!13}{711214892241391}a^{13}+\frac{10\!\cdots\!69}{711214892241391}a^{12}+\frac{41\!\cdots\!64}{711214892241391}a^{11}-\frac{10\!\cdots\!29}{14\!\cdots\!82}a^{10}-\frac{11\!\cdots\!15}{14\!\cdots\!82}a^{9}+\frac{93\!\cdots\!95}{14\!\cdots\!82}a^{8}+\frac{17\!\cdots\!13}{711214892241391}a^{7}-\frac{57\!\cdots\!95}{14\!\cdots\!82}a^{6}+\frac{47\!\cdots\!29}{14\!\cdots\!82}a^{5}-\frac{22\!\cdots\!11}{14\!\cdots\!82}a^{4}-\frac{20\!\cdots\!21}{711214892241391}a^{3}+\frac{15\!\cdots\!29}{14\!\cdots\!82}a^{2}+\frac{92\!\cdots\!69}{14\!\cdots\!82}a-\frac{46\!\cdots\!63}{14\!\cdots\!82}$, $\frac{21\!\cdots\!31}{14\!\cdots\!82}a^{24}-\frac{87\!\cdots\!97}{14\!\cdots\!82}a^{23}+\frac{52\!\cdots\!05}{711214892241391}a^{22}+\frac{54\!\cdots\!23}{14\!\cdots\!82}a^{21}-\frac{19\!\cdots\!65}{711214892241391}a^{20}+\frac{45\!\cdots\!69}{14\!\cdots\!82}a^{19}+\frac{17\!\cdots\!71}{14\!\cdots\!82}a^{18}-\frac{40\!\cdots\!97}{14\!\cdots\!82}a^{17}+\frac{14\!\cdots\!21}{14\!\cdots\!82}a^{16}-\frac{14\!\cdots\!23}{711214892241391}a^{15}+\frac{12\!\cdots\!17}{14\!\cdots\!82}a^{14}-\frac{70\!\cdots\!39}{14\!\cdots\!82}a^{13}-\frac{50\!\cdots\!09}{14\!\cdots\!82}a^{12}+\frac{22\!\cdots\!73}{14\!\cdots\!82}a^{11}+\frac{79\!\cdots\!19}{14\!\cdots\!82}a^{10}-\frac{67\!\cdots\!53}{711214892241391}a^{9}-\frac{19\!\cdots\!21}{14\!\cdots\!82}a^{8}-\frac{46\!\cdots\!37}{14\!\cdots\!82}a^{7}-\frac{58\!\cdots\!89}{14\!\cdots\!82}a^{6}+\frac{58\!\cdots\!37}{14\!\cdots\!82}a^{5}+\frac{17\!\cdots\!33}{711214892241391}a^{4}-\frac{16\!\cdots\!57}{14\!\cdots\!82}a^{3}-\frac{480134777462747}{711214892241391}a^{2}+\frac{20\!\cdots\!37}{14\!\cdots\!82}a-\frac{177556910230591}{14\!\cdots\!82}$
| 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: | \( 40076.42524869914 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 40076.42524869914 \cdot 1}{2\cdot\sqrt{190035222333323626806494766049}}\cr\approx \mathstrut & 0.141055683863403 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 4*x^24 + 4*x^23 + 5*x^22 - 20*x^21 + 18*x^20 + 19*x^19 - 28*x^18 - 2*x^16 + 2*x^15 - 28*x^14 - 29*x^13 + 120*x^12 + 59*x^11 - 112*x^10 - 35*x^9 + 16*x^8 - 23*x^7 + 29*x^6 + 32*x^5 - 21*x^4 - 12*x^3 + 7*x^2 + 2*x - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_5\times D_5$ (as 25T3):
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 50 |
| The 20 conjugacy class representatives for $C_5\times D_5$ |
| Character table for $C_5\times D_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 5.1.717409.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 10 siblings: | data not computed |
| Minimal sibling: | 10.0.246071287.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{5}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.5.0.1}{5} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{5}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.5.0.1}{5} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.5.0.1}{5} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 7.10.5.2 | $x^{10} + 35 x^{8} + 492 x^{6} + 8 x^{5} + 3360 x^{4} - 560 x^{3} + 11516 x^{2} + 1968 x + 17516$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| Deg $25$ | $5$ | $5$ | $20$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.77.10t1.a.b | $1$ | $ 7 \cdot 11 $ | 10.0.3602729712967.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.77.10t1.a.d | $1$ | $ 7 \cdot 11 $ | 10.0.3602729712967.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.77.10t1.a.a | $1$ | $ 7 \cdot 11 $ | 10.0.3602729712967.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.77.10t1.a.c | $1$ | $ 7 \cdot 11 $ | 10.0.3602729712967.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 2.77.10t6.a.c | $2$ | $ 7 \cdot 11 $ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.847.10t6.a.a | $2$ | $ 7 \cdot 11^{2}$ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.847.5t2.a.b | $2$ | $ 7 \cdot 11^{2}$ | 5.1.717409.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.847.10t6.a.c | $2$ | $ 7 \cdot 11^{2}$ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.847.10t6.a.b | $2$ | $ 7 \cdot 11^{2}$ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.847.10t6.a.d | $2$ | $ 7 \cdot 11^{2}$ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.77.10t6.a.b | $2$ | $ 7 \cdot 11 $ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.77.10t6.a.a | $2$ | $ 7 \cdot 11 $ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.847.5t2.a.a | $2$ | $ 7 \cdot 11^{2}$ | 5.1.717409.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.77.10t6.a.d | $2$ | $ 7 \cdot 11 $ | 25.5.190035222333323626806494766049.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.