Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 - y^18 + 15*y^17 - 23*y^16 - 39*y^15 + 68*y^14 + 25*y^13 - 74*y^12 + 12*y^11 + 59*y^10 - 20*y^9 - 85*y^8 + 52*y^7 + 47*y^6 - 35*y^5 - 3*y^4 - 4*y^3 + 9*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*x + 1)
\( x^{20} - 2 x^{19} - x^{18} + 15 x^{17} - 23 x^{16} - 39 x^{15} + 68 x^{14} + 25 x^{13} - 74 x^{12} + \cdots + 1 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(223904745130829775390625\)
\(\medspace = 5^{10}\cdot 13^{6}\cdot 41^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.71\) | 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: | $5^{1/2}13^{3/4}41^{3/4}\approx 248.04496106286658$ | ||
| Ramified primes: |
\(5\), \(13\), \(41\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{74883948405902}a^{19}+\frac{11331881582141}{74883948405902}a^{18}+\frac{4751088953990}{37441974202951}a^{17}+\frac{7108834316946}{37441974202951}a^{16}-\frac{16333175524573}{74883948405902}a^{15}+\frac{492831809560}{37441974202951}a^{14}+\frac{10217425025518}{37441974202951}a^{13}-\frac{23393446245307}{74883948405902}a^{12}-\frac{18400481728583}{74883948405902}a^{11}-\frac{24995338949943}{74883948405902}a^{10}-\frac{34452405634189}{74883948405902}a^{9}+\frac{8613281320819}{74883948405902}a^{8}+\frac{24890334694565}{74883948405902}a^{7}-\frac{13762020465061}{74883948405902}a^{6}+\frac{4612753089010}{37441974202951}a^{5}+\frac{3278789125636}{37441974202951}a^{4}-\frac{1337933920352}{37441974202951}a^{3}+\frac{35260833376729}{74883948405902}a^{2}+\frac{13420362032705}{74883948405902}a+\frac{25052437071387}{74883948405902}$
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: | $11$ | 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: |
$\frac{8734374628154}{37441974202951}a^{19}-\frac{7107897902531}{37441974202951}a^{18}-\frac{15390239883760}{37441974202951}a^{17}+\frac{113941969412072}{37441974202951}a^{16}-\frac{76309281337457}{37441974202951}a^{15}-\frac{414141982242317}{37441974202951}a^{14}+\frac{133495150693733}{37441974202951}a^{13}+\frac{217438327522574}{37441974202951}a^{12}-\frac{510942807432037}{37441974202951}a^{11}-\frac{148641426547683}{37441974202951}a^{10}+\frac{560591579187576}{37441974202951}a^{9}+\frac{198813036364833}{37441974202951}a^{8}-\frac{518201045676818}{37441974202951}a^{7}+\frac{139588496700101}{37441974202951}a^{6}+\frac{378291106456982}{37441974202951}a^{5}-\frac{294036526809388}{37441974202951}a^{4}-\frac{194337503062501}{37441974202951}a^{3}+\frac{48442365781022}{37441974202951}a^{2}+\frac{100754599140966}{37441974202951}a-\frac{6393563171919}{37441974202951}$, $\frac{18575942937850}{37441974202951}a^{19}-\frac{35390502361232}{37441974202951}a^{18}-\frac{20553789140907}{37441974202951}a^{17}+\frac{279270605411173}{37441974202951}a^{16}-\frac{405959452654814}{37441974202951}a^{15}-\frac{752020357943840}{37441974202951}a^{14}+\frac{12\!\cdots\!67}{37441974202951}a^{13}+\frac{488852333981078}{37441974202951}a^{12}-\frac{14\!\cdots\!87}{37441974202951}a^{11}+\frac{135749354174450}{37441974202951}a^{10}+\frac{11\!\cdots\!15}{37441974202951}a^{9}-\frac{382245030544616}{37441974202951}a^{8}-\frac{16\!\cdots\!85}{37441974202951}a^{7}+\frac{952905191009943}{37441974202951}a^{6}+\frac{968894821754032}{37441974202951}a^{5}-\frac{720729098967381}{37441974202951}a^{4}-\frac{121280334885688}{37441974202951}a^{3}-\frac{44835903377089}{37441974202951}a^{2}+\frac{185125750463204}{37441974202951}a-\frac{55308632645115}{37441974202951}$, $\frac{94960941557527}{37441974202951}a^{19}-\frac{223984063899097}{74883948405902}a^{18}-\frac{186154340812042}{37441974202951}a^{17}+\frac{12\!\cdots\!88}{37441974202951}a^{16}-\frac{22\!\cdots\!65}{74883948405902}a^{15}-\frac{46\!\cdots\!29}{37441974202951}a^{14}+\frac{26\!\cdots\!75}{37441974202951}a^{13}+\frac{45\!\cdots\!17}{37441974202951}a^{12}-\frac{68\!\cdots\!33}{74883948405902}a^{11}-\frac{16\!\cdots\!84}{37441974202951}a^{10}+\frac{87\!\cdots\!87}{74883948405902}a^{9}+\frac{33\!\cdots\!77}{74883948405902}a^{8}-\frac{67\!\cdots\!22}{37441974202951}a^{7}-\frac{492419460553862}{37441974202951}a^{6}+\frac{82\!\cdots\!25}{74883948405902}a^{5}-\frac{136543952876767}{74883948405902}a^{4}-\frac{456225982757501}{37441974202951}a^{3}-\frac{12\!\cdots\!81}{74883948405902}a^{2}+\frac{716793145741597}{74883948405902}a-\frac{228565193616367}{74883948405902}$, $\frac{21254887056633}{74883948405902}a^{19}-\frac{20496302238844}{37441974202951}a^{18}-\frac{12535221921261}{37441974202951}a^{17}+\frac{319218534777721}{74883948405902}a^{16}-\frac{463289085359137}{74883948405902}a^{15}-\frac{436142118351969}{37441974202951}a^{14}+\frac{700057189265109}{37441974202951}a^{13}+\frac{683018520195565}{74883948405902}a^{12}-\frac{779319975498908}{37441974202951}a^{11}-\frac{20385140349355}{74883948405902}a^{10}+\frac{11\!\cdots\!95}{74883948405902}a^{9}-\frac{323087067931125}{74883948405902}a^{8}-\frac{913393688139250}{37441974202951}a^{7}+\frac{449930647433025}{37441974202951}a^{6}+\frac{11\!\cdots\!29}{74883948405902}a^{5}-\frac{248703563894277}{37441974202951}a^{4}-\frac{65258538561021}{37441974202951}a^{3}-\frac{105456447821003}{37441974202951}a^{2}+\frac{102729034614836}{37441974202951}a-\frac{68899634745671}{74883948405902}$, $\frac{33823766151090}{37441974202951}a^{19}-\frac{44306016284661}{37441974202951}a^{18}-\frac{65289154732276}{37441974202951}a^{17}+\frac{928537701852361}{74883948405902}a^{16}-\frac{462533847871772}{37441974202951}a^{15}-\frac{16\!\cdots\!52}{37441974202951}a^{14}+\frac{11\!\cdots\!86}{37441974202951}a^{13}+\frac{16\!\cdots\!29}{37441974202951}a^{12}-\frac{13\!\cdots\!87}{37441974202951}a^{11}-\frac{304493477038931}{37441974202951}a^{10}+\frac{35\!\cdots\!37}{74883948405902}a^{9}+\frac{306499927314356}{37441974202951}a^{8}-\frac{50\!\cdots\!85}{74883948405902}a^{7}+\frac{108121358418898}{37441974202951}a^{6}+\frac{14\!\cdots\!58}{37441974202951}a^{5}-\frac{653865722332559}{74883948405902}a^{4}-\frac{112725317295339}{74883948405902}a^{3}-\frac{197586754366087}{74883948405902}a^{2}+\frac{190261768472311}{74883948405902}a-\frac{57940445534013}{37441974202951}$, $\frac{156429970}{114599929}a^{19}-\frac{423172891}{229199858}a^{18}-\frac{629227141}{229199858}a^{17}+\frac{2162525616}{114599929}a^{16}-\frac{2146503204}{114599929}a^{15}-\frac{7773394574}{114599929}a^{14}+\frac{5789750824}{114599929}a^{13}+\frac{8814327765}{114599929}a^{12}-\frac{12636077561}{229199858}a^{11}-\frac{6941664355}{229199858}a^{10}+\frac{15652498861}{229199858}a^{9}+\frac{4918121987}{229199858}a^{8}-\frac{12987354435}{114599929}a^{7}-\frac{1538188151}{229199858}a^{6}+\frac{8424189734}{114599929}a^{5}+\frac{250626506}{114599929}a^{4}-\frac{1391838107}{114599929}a^{3}-\frac{2714575461}{229199858}a^{2}+\frac{697875566}{114599929}a-\frac{205607056}{114599929}$, $\frac{30621015068359}{74883948405902}a^{19}-\frac{10289748194771}{74883948405902}a^{18}-\frac{85880566505927}{74883948405902}a^{17}+\frac{177994018067569}{37441974202951}a^{16}-\frac{15405563387656}{37441974202951}a^{15}-\frac{875716424361228}{37441974202951}a^{14}-\frac{219885903312433}{37441974202951}a^{13}+\frac{19\!\cdots\!39}{74883948405902}a^{12}+\frac{135525603060963}{74883948405902}a^{11}-\frac{656244239168259}{37441974202951}a^{10}+\frac{10\!\cdots\!49}{74883948405902}a^{9}+\frac{17\!\cdots\!15}{74883948405902}a^{8}-\frac{16\!\cdots\!69}{74883948405902}a^{7}-\frac{871951885169335}{37441974202951}a^{6}+\frac{10\!\cdots\!89}{74883948405902}a^{5}+\frac{905438208170361}{74883948405902}a^{4}-\frac{85851008942803}{37441974202951}a^{3}-\frac{291376346524877}{74883948405902}a^{2}-\frac{5364152240066}{37441974202951}a-\frac{21291893465139}{37441974202951}$, $\frac{6964356898460}{37441974202951}a^{19}-\frac{34683631847351}{37441974202951}a^{18}+\frac{45928022434491}{74883948405902}a^{17}+\frac{294293139086903}{74883948405902}a^{16}-\frac{906064526492495}{74883948405902}a^{15}+\frac{30928912870064}{37441974202951}a^{14}+\frac{15\!\cdots\!09}{37441974202951}a^{13}-\frac{704963368391153}{37441974202951}a^{12}-\frac{17\!\cdots\!68}{37441974202951}a^{11}+\frac{20\!\cdots\!91}{74883948405902}a^{10}+\frac{19\!\cdots\!81}{74883948405902}a^{9}-\frac{12\!\cdots\!14}{37441974202951}a^{8}-\frac{17\!\cdots\!09}{74883948405902}a^{7}+\frac{44\!\cdots\!85}{74883948405902}a^{6}+\frac{834902599276779}{74883948405902}a^{5}-\frac{15\!\cdots\!04}{37441974202951}a^{4}-\frac{252676010164327}{74883948405902}a^{3}+\frac{430377741979009}{74883948405902}a^{2}+\frac{303437117216692}{37441974202951}a-\frac{270025430643809}{74883948405902}$, $\frac{82900235229436}{37441974202951}a^{19}-\frac{166359163408581}{74883948405902}a^{18}-\frac{342116381539839}{74883948405902}a^{17}+\frac{21\!\cdots\!65}{74883948405902}a^{16}-\frac{821810534376360}{37441974202951}a^{15}-\frac{41\!\cdots\!91}{37441974202951}a^{14}+\frac{15\!\cdots\!87}{37441974202951}a^{13}+\frac{39\!\cdots\!52}{37441974202951}a^{12}-\frac{44\!\cdots\!57}{74883948405902}a^{11}-\frac{29\!\cdots\!73}{74883948405902}a^{10}+\frac{34\!\cdots\!81}{37441974202951}a^{9}+\frac{39\!\cdots\!63}{74883948405902}a^{8}-\frac{10\!\cdots\!73}{74883948405902}a^{7}-\frac{23\!\cdots\!71}{74883948405902}a^{6}+\frac{29\!\cdots\!94}{37441974202951}a^{5}+\frac{557997097455067}{74883948405902}a^{4}-\frac{205629859437137}{74883948405902}a^{3}-\frac{567142017038680}{37441974202951}a^{2}+\frac{408181516552321}{74883948405902}a-\frac{108690328090511}{37441974202951}$, $\frac{8694211722340}{37441974202951}a^{19}-\frac{2231835740895}{74883948405902}a^{18}-\frac{27585066024429}{37441974202951}a^{17}+\frac{95969526299562}{37441974202951}a^{16}+\frac{31749275952937}{74883948405902}a^{15}-\frac{522716916715891}{37441974202951}a^{14}-\frac{230753362525035}{37441974202951}a^{13}+\frac{635780522225863}{37441974202951}a^{12}+\frac{574545894680425}{74883948405902}a^{11}-\frac{355962370219803}{37441974202951}a^{10}+\frac{207463632229503}{74883948405902}a^{9}+\frac{11\!\cdots\!95}{74883948405902}a^{8}-\frac{345016161065937}{37441974202951}a^{7}-\frac{792135555789443}{37441974202951}a^{6}+\frac{326757175961471}{74883948405902}a^{5}+\frac{905430919105401}{74883948405902}a^{4}+\frac{110364754455588}{37441974202951}a^{3}-\frac{386663263029283}{74883948405902}a^{2}-\frac{19417105327245}{74883948405902}a+\frac{25113251232265}{74883948405902}$, $\frac{74161543403932}{37441974202951}a^{19}-\frac{149215290829861}{74883948405902}a^{18}-\frac{315350011422689}{74883948405902}a^{17}+\frac{19\!\cdots\!03}{74883948405902}a^{16}-\frac{14\!\cdots\!89}{74883948405902}a^{15}-\frac{37\!\cdots\!34}{37441974202951}a^{14}+\frac{14\!\cdots\!81}{37441974202951}a^{13}+\frac{37\!\cdots\!28}{37441974202951}a^{12}-\frac{37\!\cdots\!47}{74883948405902}a^{11}-\frac{28\!\cdots\!57}{74883948405902}a^{10}+\frac{30\!\cdots\!63}{37441974202951}a^{9}+\frac{17\!\cdots\!41}{37441974202951}a^{8}-\frac{96\!\cdots\!51}{74883948405902}a^{7}-\frac{12\!\cdots\!40}{37441974202951}a^{6}+\frac{27\!\cdots\!63}{37441974202951}a^{5}+\frac{817365431886321}{74883948405902}a^{4}-\frac{47342250314151}{37441974202951}a^{3}-\frac{950646707386467}{74883948405902}a^{2}+\frac{82479599984726}{37441974202951}a-\frac{150034581345441}{74883948405902}$
| 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: | \( 4108.62037976 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 4108.62037976 \cdot 1}{2\cdot\sqrt{223904745130829775390625}}\cr\approx \mathstrut & 0.168730386421 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*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^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*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^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*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^20 - 2*x^19 - x^18 + 15*x^17 - 23*x^16 - 39*x^15 + 68*x^14 + 25*x^13 - 74*x^12 + 12*x^11 + 59*x^10 - 20*x^9 - 85*x^8 + 52*x^7 + 47*x^6 - 35*x^5 - 3*x^4 - 4*x^3 + 9*x^2 - 4*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
$C_2\wr S_5$ (as 20T288):
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 non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1, 10.2.3785485925.1, 10.2.473185740625.1 |
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 |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.3785485925.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(13\)
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |