Normalized defining polynomial
\( x^{26} - x^{25} + 81 x^{24} - 82 x^{23} + 2608 x^{22} - 2691 x^{21} + 42855 x^{20} - 45630 x^{19} + 381495 x^{18} - 420657 x^{17} + 1758861 x^{16} - 2126862 x^{15} + 3181448 x^{14} - 8194220 x^{13} + 820084 x^{12} - 40536072 x^{11} + 48560594 x^{10} - 137420874 x^{9} + 249867510 x^{8} - 30122664 x^{7} + 208213844 x^{6} - 42007517 x^{5} + 1289208265 x^{4} - 2138650018 x^{3} + 1674849805 x^{2} + 2407138829 x + 1042183837 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 13]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1239285191389098448280715549883696172642141844074432051=-\,7^{13}\cdot 53^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $120.37$ | 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: | $7, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(371=7\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{371}(1,·)$, $\chi_{371}(6,·)$, $\chi_{371}(335,·)$, $\chi_{371}(356,·)$, $\chi_{371}(202,·)$, $\chi_{371}(15,·)$, $\chi_{371}(272,·)$, $\chi_{371}(146,·)$, $\chi_{371}(237,·)$, $\chi_{371}(148,·)$, $\chi_{371}(216,·)$, $\chi_{371}(281,·)$, $\chi_{371}(90,·)$, $\chi_{371}(155,·)$, $\chi_{371}(223,·)$, $\chi_{371}(225,·)$, $\chi_{371}(99,·)$, $\chi_{371}(36,·)$, $\chi_{371}(134,·)$, $\chi_{371}(169,·)$, $\chi_{371}(365,·)$, $\chi_{371}(370,·)$, $\chi_{371}(309,·)$, $\chi_{371}(183,·)$, $\chi_{371}(188,·)$, $\chi_{371}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{8881} a^{23} - \frac{1188}{8881} a^{22} + \frac{84}{8881} a^{21} + \frac{1127}{8881} a^{20} - \frac{2311}{8881} a^{19} + \frac{1219}{8881} a^{18} - \frac{2235}{8881} a^{17} - \frac{1941}{8881} a^{16} - \frac{1064}{8881} a^{15} - \frac{1085}{8881} a^{14} - \frac{1593}{8881} a^{13} + \frac{1879}{8881} a^{12} - \frac{2963}{8881} a^{11} - \frac{191}{8881} a^{10} - \frac{2872}{8881} a^{9} - \frac{2295}{8881} a^{8} + \frac{4200}{8881} a^{7} + \frac{1582}{8881} a^{6} + \frac{2926}{8881} a^{5} - \frac{2102}{8881} a^{4} + \frac{1903}{8881} a^{3} + \frac{221}{8881} a^{2} + \frac{1220}{8881} a + \frac{3980}{8881}$, $\frac{1}{8881} a^{24} + \frac{819}{8881} a^{22} + \frac{3228}{8881} a^{21} + \frac{4415}{8881} a^{20} - \frac{20}{8881} a^{19} - \frac{1666}{8881} a^{18} - \frac{1702}{8881} a^{17} + \frac{2088}{8881} a^{16} - \frac{4015}{8881} a^{15} - \frac{2828}{8881} a^{14} + \frac{1048}{8881} a^{13} + \frac{158}{8881} a^{12} - \frac{3359}{8881} a^{11} + \frac{1126}{8881} a^{10} - \frac{3927}{8881} a^{9} + \frac{4207}{8881} a^{8} + \frac{60}{8881} a^{7} - \frac{430}{8881} a^{6} + \frac{1515}{8881} a^{5} + \frac{288}{8881} a^{4} - \frac{3670}{8881} a^{3} - \frac{2662}{8881} a^{2} - \frac{3144}{8881} a + \frac{3548}{8881}$, $\frac{1}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{25} - \frac{1036313208294455188951908582793312763538400887290821468673339095345016536413511956670857289331833911714127134084063255}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{24} - \frac{410010259837394855651072321030452970125876668115674011197659919872335757280343754386670115936111103940016917786676051}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{23} - \frac{1796514873580983931339440867282406224259696657660474100423975519460007267959045873770564902157315330947793191029822775307}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{22} + \frac{9555322140109051645279430209448244859605746943257782871113089062528794956079569018131627280248926554958808114917650838787}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{21} + \frac{18930070110098413867201592990030079890832609007417334169660347689175717485331813519027141733573299042349442727094795547542}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{20} + \frac{3061455225311947289274742995019476553049490894104835044803032913372510227138081782532637406768668761779592519696610270048}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{19} + \frac{44892699682557720898677185917113466049886985877808701599841360286695656248189663371857991642565129905661042733641585251}{360200000382390777198547781733186592200256517358294030248122790477558161974183640580795258648990584498492155892804124317} a^{18} - \frac{1864887092306774295233342376551865984605532805175968215973806642045807948852231809883650942305129295339782779966751899264}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{17} + \frac{15898340015974975741188156701493434834980317064221546816431112691294175424820791471413402748596243807396940623671456167643}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{16} - \frac{16211556329687648850727650602093110402798060732278663981728895006927228805550019272652030311502493809301610411343812972103}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{15} - \frac{11327242022555748415934680760196546442615380477759591107385739079334360736788563672165128362576045709476932719250306960930}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{14} - \frac{12228153669759992251248364019642829356881990712637138240571008852421783390024449457152219922724721752560198061848092676759}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{13} - \frac{3134055621446034701121122339571218161989311314613782061042135843666630167243061048357495211405660152393401119981361657377}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{12} - \frac{12112467680418117076946195364893855750262705617982071564988597110272002797287045286738852964854363364868049823163900270544}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{11} + \frac{18121213630219698285722282488854238257292110151316509247254886595338988967441831638686409538230795527877067433731864646252}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{10} - \frac{18541504946198960354008223783995045990742445068752525483652027965008000928574222927338383635189185693999452779485893626308}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{9} - \frac{362156160705730492069143703700661285415293810808177000429558763316918961494370908657215260059200013423958312169774992153}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{8} - \frac{587656196225022308831413497007428361619834760520658754812844483575544429032251834157016912470361041779989896221535350901}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{7} + \frac{1044771855892972385557629458156789076229129802968439124304980308011579819949107315527156783567854193682139384914701492700}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{6} - \frac{17855232584241185180913602745005722887825165664604255076221648940411192543800042825938601417495945688920847192242802285571}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{5} + \frac{2495163929619470377379653382605034304393745184315644782445053831366607799524646375380903929129958776189195956376902416581}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{4} - \frac{18536629447368735468778088378906118196211652418565249534591712984476272272165658451775576078438711754685971697736902292251}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{3} - \frac{2822100663115909760296486276607676770177948092447137753332208552728794365350805049480351283864564789956561989230911853898}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a^{2} + \frac{6696661027195543703069573151528261058623267077093188026303334169197237940860609678174831969117338190874425084591617203459}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919} a - \frac{17041802081842532569845553396378924960038488134176662541455123138262647558254773353606030513737956543066294860598543556675}{38541400040915813160244612645450965365427447357337461236549138581098723331237649542145092675441992541338660680530041301919}$
Class group and class number
$C_{53}\times C_{66568}$, which has order $3528104$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 5382739421.971964 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-371}) \), 13.13.491258904256726154641.1 |
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 | $26$ | ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ | R | ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | R | $26$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 53 | Data not computed | ||||||