Normalized defining polynomial
\( x^{21} - 2 x^{20} - 428 x^{19} - 260 x^{18} + 60792 x^{17} + 119542 x^{16} - 3787421 x^{15} - 11196620 x^{14} + 115772376 x^{13} + 456080942 x^{12} - 1718615710 x^{11} - 9374121523 x^{10} + 9670206925 x^{9} + 102252308673 x^{8} + 33867547395 x^{7} - 577620233371 x^{6} - 647428833299 x^{5} + 1458253090825 x^{4} + 2486759937972 x^{3} - 984522959681 x^{2} - 2347418153468 x + 261660725120 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[17, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(673490452509783500077001584367520548569368129410325212007271089=3^{6}\cdot 7^{12}\cdot 173^{10}\cdot 859^{2}\cdot 61374792667^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $981.35$ | 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: | $3, 7, 173, 859, 61374792667$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| 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}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{20} - \frac{19825413053971740179438528254346864476036696587027091900066582710544085637149474466120477606800093967980988298909534455}{187037778654847228594767411645453140163010780301970484228689933764091525437917626631017710583360393771923671422639498022} a^{19} + \frac{26470085761114870279316878582948833304803613739877346162217931200887489519645333594729824130772420083213623977306581519}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{18} + \frac{12572146057984437272193954701761994092518689088249893184648410068053852684142428121610647686743374393997000520918841991}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{17} + \frac{31003508266165064584603051415335801809842100444639724641132520526881089621882053629912039894427784852734151740731151737}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{16} - \frac{37155511963394816569485900141232741263312528284833260904168123999579369062865579510762721505715845693911226226100455843}{187037778654847228594767411645453140163010780301970484228689933764091525437917626631017710583360393771923671422639498022} a^{15} - \frac{31145070692110776442271163579030316952785333001220059121902783648092978731378224177802021009798662773527197004542944641}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{14} - \frac{18762224156888604393143383705368573952803464001244136356086036089194718402742099250700943220117438524539216995504501454}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{13} + \frac{26088170271555350505688146602092112419258967190239975685094005449738456582182417054046499183529613479957846641526277874}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{12} - \frac{41384337644869231504768654701037823275391837915741910136731148414055031566947257230855554758251452244939598589921247095}{187037778654847228594767411645453140163010780301970484228689933764091525437917626631017710583360393771923671422639498022} a^{11} + \frac{22243490820606563968102674869984557682134439182052208437110427476844304865087188950545848642786376490689912653014854693}{187037778654847228594767411645453140163010780301970484228689933764091525437917626631017710583360393771923671422639498022} a^{10} - \frac{36833354324855635008242415616283840527279360213698251613456277527678930272847638583112747730777752542214881693421728967}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{9} - \frac{41076657373822828093346995523242357001306201358849511475818786877775006989155023217051387573485683273286466353324207435}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{8} + \frac{5748869393422530101654534566553468861395813735940186448069306287333239542530455694236445970659037568717927451076950053}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{7} - \frac{114051192996736288155890482360044572810672210800838337630615985018180179834299821798290250931138393249814458298143334977}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{6} - \frac{82923394812121446940117513788750068210130888164217550083099360462482851282267685294085496411901666809403180998989214767}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{5} - \frac{56173527271018830946473592256291470476844091485734903886830245329209718435442991876562428047700731098501018730538961171}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{4} + \frac{38247645954222137184760388431574781080978320302352427744688259708384098951486735548515924236729253037071910997374034397}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a^{3} + \frac{41928013487874405061391868515362832760710626470575264942022217179923511921406193815226782064618428309844675669164003535}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011} a^{2} - \frac{130786828110415138603383842915731554849976863330961709398827053663461220353884326619554803008304762585184032592263360469}{374075557309694457189534823290906280326021560603940968457379867528183050875835253262035421166720787543847342845278996044} a + \frac{27741319926451611311797709994288885239022562038798003574580441847367204835526453468477895374800233413717915023230978627}{93518889327423614297383705822726570081505390150985242114344966882045762718958813315508855291680196885961835711319749011}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $18$ | 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: | \( 17865297648200000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5878656 |
| The 96 conjugacy class representatives for t21n134 are not computed |
| Character table for t21n134 is not computed |
Intermediate fields
| 7.7.12431698517.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ | R | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | $18{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ | $21$ | ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ | $21$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | $18{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 7 | Data not computed | ||||||
| 173 | Data not computed | ||||||
| 859 | Data not computed | ||||||
| 61374792667 | Data not computed | ||||||