Normalized defining polynomial
\( x^{25} - 550 x^{23} + 124025 x^{21} - 14973750 x^{19} - 468875 x^{18} + 1062231775 x^{17} + 127909100 x^{16} - 45741591885 x^{15} - 12878589625 x^{14} + 1194292723800 x^{13} + 607824841800 x^{12} - 18393612016450 x^{11} - 14200548281755 x^{10} + 158788744054875 x^{9} + 159805347014225 x^{8} - 710266705493675 x^{7} - 819704665921625 x^{6} + 1448128599562260 x^{5} + 1723030958622875 x^{4} - 1001465382017100 x^{3} - 1173337004132150 x^{2} - 200208626090550 x - 9441969348251 \)
Invariants
| Degree: | $25$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[25, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(227936564389216769066972959924266550757465665810741484165191650390625=5^{68}\cdot 11^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $542.39$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1375=5^{3}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1375}(896,·)$, $\chi_{1375}(1,·)$, $\chi_{1375}(961,·)$, $\chi_{1375}(581,·)$, $\chi_{1375}(641,·)$, $\chi_{1375}(136,·)$, $\chi_{1375}(1101,·)$, $\chi_{1375}(1171,·)$, $\chi_{1375}(276,·)$, $\chi_{1375}(366,·)$, $\chi_{1375}(856,·)$, $\chi_{1375}(71,·)$, $\chi_{1375}(411,·)$, $\chi_{1375}(346,·)$, $\chi_{1375}(31,·)$, $\chi_{1375}(91,·)$, $\chi_{1375}(1191,·)$, $\chi_{1375}(551,·)$, $\chi_{1375}(1131,·)$, $\chi_{1375}(621,·)$, $\chi_{1375}(686,·)$, $\chi_{1375}(916,·)$, $\chi_{1375}(306,·)$, $\chi_{1375}(1236,·)$, $\chi_{1375}(826,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{77} a^{6} - \frac{2}{7}$, $\frac{1}{77} a^{7} - \frac{2}{7} a$, $\frac{1}{77} a^{8} - \frac{2}{7} a^{2}$, $\frac{1}{77} a^{9} - \frac{2}{7} a^{3}$, $\frac{1}{847} a^{10} + \frac{3}{7} a^{4}$, $\frac{1}{5929} a^{11} - \frac{1}{5929} a^{10} + \frac{2}{539} a^{9} - \frac{2}{539} a^{8} + \frac{2}{539} a^{7} - \frac{2}{539} a^{6} + \frac{19}{539} a^{5} - \frac{24}{49} a^{4} - \frac{11}{49} a^{3} + \frac{11}{49} a^{2} - \frac{11}{49} a + \frac{11}{49}$, $\frac{1}{5929} a^{12} + \frac{3}{539} a^{6} - \frac{10}{49}$, $\frac{1}{5929} a^{13} + \frac{3}{539} a^{7} - \frac{10}{49} a$, $\frac{1}{5929} a^{14} + \frac{3}{539} a^{8} - \frac{10}{49} a^{2}$, $\frac{1}{65219} a^{15} - \frac{1}{539} a^{9} + \frac{15}{49} a^{3}$, $\frac{1}{456533} a^{16} - \frac{2}{456533} a^{15} - \frac{1}{41503} a^{14} - \frac{1}{41503} a^{13} + \frac{3}{41503} a^{12} + \frac{2}{41503} a^{11} + \frac{8}{41503} a^{10} - \frac{15}{3773} a^{9} + \frac{3}{539} a^{8} + \frac{8}{3773} a^{7} + \frac{12}{3773} a^{6} - \frac{158}{3773} a^{5} + \frac{30}{343} a^{4} + \frac{137}{343} a^{3} + \frac{123}{343} a^{2} + \frac{121}{343} a - \frac{120}{343}$, $\frac{1}{456533} a^{17} - \frac{1}{456533} a^{15} - \frac{3}{41503} a^{14} + \frac{1}{41503} a^{13} + \frac{1}{41503} a^{12} - \frac{2}{41503} a^{11} + \frac{12}{41503} a^{10} - \frac{2}{3773} a^{9} - \frac{20}{3773} a^{8} + \frac{20}{3773} a^{6} + \frac{13}{539} a^{5} - \frac{55}{343} a^{4} - \frac{23}{343} a^{3} + \frac{66}{343} a^{2} - \frac{67}{343} a + \frac{68}{343}$, $\frac{1}{456533} a^{18} + \frac{1}{41503} a^{12} - \frac{16}{3773} a^{6} + \frac{20}{343}$, $\frac{1}{3195731} a^{19} + \frac{2}{3195731} a^{18} - \frac{2}{3195731} a^{17} + \frac{2}{3195731} a^{15} - \frac{8}{290521} a^{14} - \frac{8}{290521} a^{13} - \frac{3}{41503} a^{12} - \frac{17}{290521} a^{11} - \frac{150}{290521} a^{10} - \frac{136}{26411} a^{9} - \frac{107}{26411} a^{8} + \frac{166}{26411} a^{7} + \frac{103}{26411} a^{6} - \frac{34}{3773} a^{5} - \frac{856}{2401} a^{4} - \frac{213}{2401} a^{3} + \frac{1100}{2401} a^{2} + \frac{142}{343} a + \frac{863}{2401}$, $\frac{1}{3761375387} a^{20} - \frac{32}{341943217} a^{19} + \frac{30}{31085747} a^{18} + \frac{1}{3195731} a^{17} - \frac{157}{341943217} a^{16} - \frac{90}{31085747} a^{15} + \frac{1511}{31085747} a^{14} + \frac{1618}{31085747} a^{13} + \frac{239}{31085747} a^{12} - \frac{2334}{31085747} a^{11} - \frac{2496}{4440821} a^{10} - \frac{3312}{2825977} a^{9} + \frac{5920}{2825977} a^{8} + \frac{16658}{2825977} a^{7} - \frac{3582}{2825977} a^{6} + \frac{90781}{2825977} a^{5} + \frac{45146}{256907} a^{4} - \frac{40336}{256907} a^{3} - \frac{25026}{256907} a^{2} + \frac{27808}{256907} a - \frac{12714}{256907}$, $\frac{1}{26329627709} a^{21} - \frac{3}{26329627709} a^{20} - \frac{32}{217600229} a^{19} - \frac{818}{2393602519} a^{18} + \frac{1234}{2393602519} a^{17} - \frac{51}{341943217} a^{16} + \frac{10408}{2393602519} a^{15} - \frac{16315}{217600229} a^{14} + \frac{239}{31085747} a^{13} + \frac{6177}{217600229} a^{12} + \frac{850}{217600229} a^{11} - \frac{56025}{217600229} a^{10} - \frac{69161}{19781839} a^{9} - \frac{6453}{2825977} a^{8} - \frac{6674}{19781839} a^{7} + \frac{30824}{19781839} a^{6} - \frac{54479}{2825977} a^{5} - \frac{504726}{1798349} a^{4} + \frac{462764}{1798349} a^{3} - \frac{602086}{1798349} a^{2} + \frac{63883}{1798349} a - \frac{728887}{1798349}$, $\frac{1}{194812915418891} a^{22} + \frac{1088}{194812915418891} a^{21} - \frac{22860}{194812915418891} a^{20} - \frac{2661509}{17710265038081} a^{19} - \frac{1161746}{1610024094371} a^{18} + \frac{881912}{1610024094371} a^{17} - \frac{12361532}{17710265038081} a^{16} + \frac{8344814}{17710265038081} a^{15} + \frac{56396632}{1610024094371} a^{14} - \frac{41850827}{1610024094371} a^{13} - \frac{1738314}{230003442053} a^{12} - \frac{8272417}{230003442053} a^{11} + \frac{6499522}{230003442053} a^{10} - \frac{748199871}{146365826761} a^{9} + \frac{511860059}{146365826761} a^{8} - \frac{530843930}{146365826761} a^{7} + \frac{877327224}{146365826761} a^{6} + \frac{3449951051}{146365826761} a^{5} + \frac{3738406818}{13305984251} a^{4} - \frac{1790416099}{13305984251} a^{3} - \frac{3425467056}{13305984251} a^{2} + \frac{3635797425}{13305984251} a - \frac{6235227917}{13305984251}$, $\frac{1}{1363690407932237} a^{23} + \frac{2}{1363690407932237} a^{22} - \frac{109}{11270168660597} a^{21} - \frac{48234}{1363690407932237} a^{20} - \frac{739183}{11270168660597} a^{19} - \frac{49163208}{123971855266567} a^{18} - \frac{38934377}{123971855266567} a^{17} + \frac{78765032}{123971855266567} a^{16} - \frac{177133491}{123971855266567} a^{15} + \frac{185761553}{11270168660597} a^{14} - \frac{44656219}{1024560787327} a^{13} + \frac{88527248}{1610024094371} a^{12} - \frac{266955}{4693947797} a^{11} - \frac{5570404549}{11270168660597} a^{10} + \frac{858827098}{1024560787327} a^{9} + \frac{4306520535}{1024560787327} a^{8} - \frac{2849832995}{1024560787327} a^{7} - \frac{6307661538}{1024560787327} a^{6} + \frac{20027594001}{1024560787327} a^{5} + \frac{19180206092}{93141889757} a^{4} - \frac{20172990685}{93141889757} a^{3} + \frac{26834589024}{93141889757} a^{2} - \frac{18691423523}{93141889757} a + \frac{36787934948}{93141889757}$, $\frac{1}{441979013141168817500256051778062572024033242221959697257228984062651059155306802391} a^{24} - \frac{107692984819386634407034464502375835222274632882044980498211423087196}{441979013141168817500256051778062572024033242221959697257228984062651059155306802391} a^{23} + \frac{99857799467348062859650594265484070981764396195263320620762927809059}{441979013141168817500256051778062572024033242221959697257228984062651059155306802391} a^{22} + \frac{5953441458513469574010235439495742854154763567664223446599401965552187667}{441979013141168817500256051778062572024033242221959697257228984062651059155306802391} a^{21} - \frac{45093374677521649058539679983998339884056045353911131197861335612809750730}{441979013141168817500256051778062572024033242221959697257228984062651059155306802391} a^{20} - \frac{4244816026663930445225119900013114795254292883954976844736652784562139925342}{40179910285560801590932368343460233820366658383814517932475362187513732650482436581} a^{19} - \frac{30068716867024100299238440531212877836617127816160489165346164177308467295825}{40179910285560801590932368343460233820366658383814517932475362187513732650482436581} a^{18} - \frac{24312947550511853053967494299147743058234720360376108844993958402870664952754}{40179910285560801590932368343460233820366658383814517932475362187513732650482436581} a^{17} + \frac{5508723361212901700127210333970378339563423397423637225425374704940776452496}{40179910285560801590932368343460233820366658383814517932475362187513732650482436581} a^{16} + \frac{16907186591043298020035093644710466070693260591608397974694618597831081343161}{3652719116869163780993851667587293983669696216710410721134123835228521150043857871} a^{15} - \frac{2420743271742313960019496982166928308926168761141383355926201224939308199825}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a^{14} - \frac{176932751772707712755822498506967593438439296196833759983019960444915089051707}{3652719116869163780993851667587293983669696216710410721134123835228521150043857871} a^{13} - \frac{11599631790808644571391579318289319155709974674604831311618875520077890457787}{521817016695594825856264523941041997667099459530058674447731976461217307149122553} a^{12} - \frac{118966484284599637427382580611008773545641145830177223109768724036692872529133}{3652719116869163780993851667587293983669696216710410721134123835228521150043857871} a^{11} - \frac{1461914935464364813040736817080696594460518679086780647658301200624748249329187}{3652719116869163780993851667587293983669696216710410721134123835228521150043857871} a^{10} - \frac{182302519047737865243250699546479617217726033493587680265570854714839832348160}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a^{9} + \frac{249461754297940772707038153007617557544176524438444086631765698081422757198}{332065374260833070999441060689753998515426928791855520103102166838956468185805261} a^{8} - \frac{726290992695959816257246327134387530752720825929783453310013990962919938015397}{332065374260833070999441060689753998515426928791855520103102166838956468185805261} a^{7} - \frac{969960383679980926639877562735766015311748622257114233667792727363601258982768}{332065374260833070999441060689753998515426928791855520103102166838956468185805261} a^{6} + \frac{3730554698828823297742256064474534368542038729952479872625153718435758192807430}{332065374260833070999441060689753998515426928791855520103102166838956468185805261} a^{5} - \frac{4241514911940433375295973698167722878549193786761029653171567997015045624801797}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a^{4} - \frac{760038791745129443556021629617660356921815131259533868811288887817236371217049}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a^{3} - \frac{10149730452613269217633465522554792252001315115625559124102664700612484535903260}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a^{2} - \frac{13375303596949650127382484085932656782108004731037346978619424541826375837587456}{30187761296439370090858278244523090774129720799259592736645651530814224380527751} a - \frac{14978553997604160632727439223526207854728366099071892274775638041575075987426490}{30187761296439370090858278244523090774129720799259592736645651530814224380527751}$
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
| Rank: | $24$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1747365352353972600000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 25 |
| The 25 conjugacy class representatives for $C_{25}$ |
| Character table for $C_{25}$ is not computed |
Intermediate fields
| 5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $25$ | $25$ | R | ${\href{/LocalNumberField/7.1.0.1}{1} }^{25}$ | R | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ | $25$ | $25$ | $25$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||