Normalized defining polynomial
\( x^{21} - 2 x^{20} - 278 x^{19} + 1209 x^{18} + 28609 x^{17} - 188663 x^{16} - 1265361 x^{15} + 12878599 x^{14} + 13360899 x^{13} - 429530466 x^{12} + 780621370 x^{11} + 6395176809 x^{10} - 27439148621 x^{9} - 14072788600 x^{8} + 293371856110 x^{7} - 536805970235 x^{6} - 490924632667 x^{5} + 3155238661458 x^{4} - 4701437594680 x^{3} + 2873237146615 x^{2} - 294214513112 x - 270275728417 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1312348190467104151476370758095143331434441717851465322827=-\,3^{7}\cdot 29^{18}\cdot 157^{2}\cdot 3307^{2}\cdot 3943^{2}\cdot 26083^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $524.70$ | 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: | $3, 29, 157, 3307, 3943, 26083$ | 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}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{20} - \frac{192430026279278107468226423304951546377815094250415736756732877945972884903004488249872407592441676337}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{19} - \frac{181210755243128577107701121583490171792714912172620338207377842586040672936775450562979284642290407410}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{18} - \frac{113429503189256499894072589182316573008221376080339847460810491087193778756584264624173472522619635441}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{17} - \frac{62591368148784230692619176821158560146529830279124185186396177876162177060149100989260846347136878855}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{16} - \frac{32350206738628179901987504083297350285292340196578687379473855654022160956713980839232539751081929062}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{15} + \frac{301736102526317248977135712631664056878165897836549889748896911464640916309281244000315947310294523656}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{14} + \frac{155430244015609771123301803073776636590706827189936655490184251763306578316537187672926744577505178210}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{13} + \frac{190782489124623575173769102003789569691488874628867467312415881842284544511878941547776651226014717204}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{12} - \frac{75447274199748290361776249679800734303704652378931664481938996470468431208360490718166250906425350499}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{11} - \frac{65478865148451018119818054605879101404696374227174247079721540241540763382884151714325112775292453883}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{10} + \frac{289264669285957244523241925483361457411496335025079908834671853755272448529582338822449326804348034180}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{9} + \frac{149118652162803328344673546959155144340148364455913469755823935754750303568207069711249224338091779981}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{8} + \frac{231674886495310776997895660254897982483291744733050107559454093427069862470913401388401029171262746684}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{7} + \frac{303631884851663148982644948134680009502008830342663544906443434630850426937761317227594696267544474024}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{6} + \frac{20499218578328425424653288768957827640075585011334634951621229323752373107754724431107352356887037998}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{5} - \frac{159697976324802145676083413474907363728958117213518803190519218894088049068092393520577306258862158923}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{4} - \frac{311897356587515324049602021258335145673640904024627055394954069556768235931243505250040384939043557972}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{3} - \frac{153559552362355326142928925040156872536979121169460893929112147282086350887084833201944246042358180397}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a^{2} + \frac{49092513605236423349425192547171055179906663634919461226199872431301583071755611649209440411369844732}{641706605269074089663642081775679140470752504024695392581372819839670035618949905027555413060316999733} a + \frac{2704751026909458639280838509361756073663033109155583071357852788974518377593645289215306496208666711}{15651380616318880235698587360382418060262256195724277867838361459504147210218290366525741781958951213}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 616510170935000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 244944 |
| The 72 conjugacy class representatives for t21n112 are not computed |
| Character table for t21n112 is not computed |
Intermediate fields
| 7.7.594823321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ | $21$ | ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | $21$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ | R | $21$ | $21$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{9}$ | $21$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.7.0.1 | $x^{7} + x^{2} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 3.14.7.2 | $x^{14} + 243 x^{4} - 729 x^{2} + 2187$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
| $29$ | 29.7.6.2 | $x^{7} - 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.14.12.1 | $x^{14} + 2407 x^{7} + 1839267$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |
| $157$ | $\Q_{157}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{157}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{157}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{157}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 157.3.0.1 | $x^{3} - x + 15$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 157.3.0.1 | $x^{3} - x + 15$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 157.3.2.3 | $x^{3} - 3925$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 3307 | Data not computed | ||||||
| 3943 | Data not computed | ||||||
| 26083 | Data not computed | ||||||