Normalized defining polynomial
\( x^{13} - x^{12} - 72 x^{11} + 129 x^{10} + 1672 x^{9} - 3386 x^{8} - 16810 x^{7} + 32367 x^{6} + \cdots + 61399 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[13, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(224282727500720205065439601\) \(\medspace = 157^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(106.41\) | 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: | $157^{12/13}\approx 106.41045361273743$ | ||
Ramified primes: | \(157\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $13$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(157\) | ||
Dirichlet character group: | $\lbrace$$\chi_{157}(1,·)$, $\chi_{157}(130,·)$, $\chi_{157}(67,·)$, $\chi_{157}(101,·)$, $\chi_{157}(39,·)$, $\chi_{157}(75,·)$, $\chi_{157}(108,·)$, $\chi_{157}(14,·)$, $\chi_{157}(16,·)$, $\chi_{157}(99,·)$, $\chi_{157}(46,·)$, $\chi_{157}(153,·)$, $\chi_{157}(93,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}+\frac{4}{13}a^{7}+\frac{2}{13}a^{6}-\frac{3}{13}a^{5}-\frac{3}{13}a^{4}+\frac{5}{13}a^{3}-\frac{6}{13}a$, $\frac{1}{13}a^{9}-\frac{1}{13}a^{7}+\frac{2}{13}a^{6}-\frac{4}{13}a^{5}+\frac{4}{13}a^{4}+\frac{6}{13}a^{3}-\frac{6}{13}a^{2}-\frac{2}{13}a$, $\frac{1}{13}a^{10}+\frac{6}{13}a^{7}-\frac{2}{13}a^{6}+\frac{1}{13}a^{5}+\frac{3}{13}a^{4}-\frac{1}{13}a^{3}-\frac{2}{13}a^{2}-\frac{6}{13}a$, $\frac{1}{13}a^{11}+\frac{2}{13}a^{6}-\frac{5}{13}a^{5}+\frac{4}{13}a^{4}-\frac{6}{13}a^{3}-\frac{6}{13}a^{2}-\frac{3}{13}a$, $\frac{1}{33\!\cdots\!81}a^{12}+\frac{95\!\cdots\!37}{33\!\cdots\!81}a^{11}+\frac{60\!\cdots\!92}{33\!\cdots\!81}a^{10}-\frac{36\!\cdots\!86}{33\!\cdots\!81}a^{9}+\frac{31\!\cdots\!74}{33\!\cdots\!81}a^{8}+\frac{65\!\cdots\!39}{33\!\cdots\!81}a^{7}+\frac{11\!\cdots\!73}{33\!\cdots\!81}a^{6}+\frac{14\!\cdots\!28}{33\!\cdots\!81}a^{5}-\frac{66\!\cdots\!53}{33\!\cdots\!81}a^{4}+\frac{47\!\cdots\!38}{25\!\cdots\!37}a^{3}-\frac{54\!\cdots\!02}{33\!\cdots\!81}a^{2}+\frac{15\!\cdots\!24}{33\!\cdots\!81}a+\frac{51\!\cdots\!59}{25\!\cdots\!37}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $\frac{184298391802910}{33\!\cdots\!81}a^{12}-\frac{147652985028021}{33\!\cdots\!81}a^{11}-\frac{11\!\cdots\!34}{33\!\cdots\!81}a^{10}+\frac{21\!\cdots\!12}{33\!\cdots\!81}a^{9}+\frac{20\!\cdots\!86}{33\!\cdots\!81}a^{8}-\frac{47\!\cdots\!49}{33\!\cdots\!81}a^{7}-\frac{12\!\cdots\!05}{33\!\cdots\!81}a^{6}+\frac{30\!\cdots\!47}{33\!\cdots\!81}a^{5}+\frac{25\!\cdots\!69}{33\!\cdots\!81}a^{4}-\frac{47\!\cdots\!53}{25\!\cdots\!37}a^{3}-\frac{31\!\cdots\!03}{33\!\cdots\!81}a^{2}+\frac{40\!\cdots\!82}{33\!\cdots\!81}a+\frac{17\!\cdots\!69}{25\!\cdots\!37}$, $\frac{40474592848696}{33\!\cdots\!81}a^{12}-\frac{782135619101090}{33\!\cdots\!81}a^{11}-\frac{19\!\cdots\!28}{33\!\cdots\!81}a^{10}+\frac{52\!\cdots\!76}{33\!\cdots\!81}a^{9}-\frac{36\!\cdots\!75}{33\!\cdots\!81}a^{8}-\frac{99\!\cdots\!62}{33\!\cdots\!81}a^{7}+\frac{15\!\cdots\!13}{33\!\cdots\!81}a^{6}+\frac{66\!\cdots\!76}{33\!\cdots\!81}a^{5}-\frac{10\!\cdots\!87}{33\!\cdots\!81}a^{4}-\frac{13\!\cdots\!61}{25\!\cdots\!37}a^{3}+\frac{13\!\cdots\!24}{33\!\cdots\!81}a^{2}+\frac{22\!\cdots\!84}{33\!\cdots\!81}a+\frac{50\!\cdots\!17}{25\!\cdots\!37}$, $\frac{178669226132154}{33\!\cdots\!81}a^{12}-\frac{112686280406228}{33\!\cdots\!81}a^{11}-\frac{11\!\cdots\!93}{33\!\cdots\!81}a^{10}+\frac{17\!\cdots\!68}{33\!\cdots\!81}a^{9}+\frac{22\!\cdots\!10}{33\!\cdots\!81}a^{8}-\frac{40\!\cdots\!09}{33\!\cdots\!81}a^{7}-\frac{16\!\cdots\!68}{33\!\cdots\!81}a^{6}+\frac{25\!\cdots\!09}{33\!\cdots\!81}a^{5}+\frac{49\!\cdots\!69}{33\!\cdots\!81}a^{4}-\frac{34\!\cdots\!15}{25\!\cdots\!37}a^{3}-\frac{66\!\cdots\!14}{33\!\cdots\!81}a^{2}+\frac{77\!\cdots\!35}{33\!\cdots\!81}a+\frac{98\!\cdots\!03}{25\!\cdots\!37}$, $\frac{53953733107084}{33\!\cdots\!81}a^{12}-\frac{302520823761437}{33\!\cdots\!81}a^{11}-\frac{45\!\cdots\!93}{33\!\cdots\!81}a^{10}+\frac{21\!\cdots\!75}{33\!\cdots\!81}a^{9}+\frac{10\!\cdots\!99}{33\!\cdots\!81}a^{8}-\frac{47\!\cdots\!42}{33\!\cdots\!81}a^{7}-\frac{10\!\cdots\!44}{33\!\cdots\!81}a^{6}+\frac{42\!\cdots\!92}{33\!\cdots\!81}a^{5}+\frac{48\!\cdots\!38}{33\!\cdots\!81}a^{4}-\frac{12\!\cdots\!41}{25\!\cdots\!37}a^{3}-\frac{10\!\cdots\!40}{33\!\cdots\!81}a^{2}+\frac{18\!\cdots\!63}{33\!\cdots\!81}a+\frac{94\!\cdots\!55}{25\!\cdots\!37}$, $\frac{75728491094842}{33\!\cdots\!81}a^{12}+\frac{315135247419604}{33\!\cdots\!81}a^{11}-\frac{50\!\cdots\!08}{33\!\cdots\!81}a^{10}-\frac{17\!\cdots\!89}{33\!\cdots\!81}a^{9}+\frac{12\!\cdots\!29}{33\!\cdots\!81}a^{8}+\frac{35\!\cdots\!32}{33\!\cdots\!81}a^{7}-\frac{12\!\cdots\!59}{33\!\cdots\!81}a^{6}-\frac{35\!\cdots\!37}{33\!\cdots\!81}a^{5}+\frac{58\!\cdots\!92}{33\!\cdots\!81}a^{4}+\frac{12\!\cdots\!13}{25\!\cdots\!37}a^{3}-\frac{72\!\cdots\!95}{33\!\cdots\!81}a^{2}-\frac{23\!\cdots\!63}{33\!\cdots\!81}a-\frac{50\!\cdots\!82}{25\!\cdots\!37}$, $\frac{88788483830118}{33\!\cdots\!81}a^{12}-\frac{215005733386385}{33\!\cdots\!81}a^{11}-\frac{59\!\cdots\!82}{33\!\cdots\!81}a^{10}+\frac{21\!\cdots\!70}{33\!\cdots\!81}a^{9}+\frac{11\!\cdots\!55}{33\!\cdots\!81}a^{8}-\frac{51\!\cdots\!50}{33\!\cdots\!81}a^{7}-\frac{73\!\cdots\!85}{33\!\cdots\!81}a^{6}+\frac{47\!\cdots\!69}{33\!\cdots\!81}a^{5}+\frac{96\!\cdots\!96}{33\!\cdots\!81}a^{4}-\frac{13\!\cdots\!64}{25\!\cdots\!37}a^{3}+\frac{19\!\cdots\!88}{33\!\cdots\!81}a^{2}+\frac{17\!\cdots\!88}{33\!\cdots\!81}a+\frac{47\!\cdots\!32}{25\!\cdots\!37}$, $\frac{183278562193037}{33\!\cdots\!81}a^{12}-\frac{82538987421593}{33\!\cdots\!81}a^{11}-\frac{12\!\cdots\!25}{33\!\cdots\!81}a^{10}+\frac{15\!\cdots\!36}{33\!\cdots\!81}a^{9}+\frac{25\!\cdots\!58}{33\!\cdots\!81}a^{8}-\frac{33\!\cdots\!73}{33\!\cdots\!81}a^{7}-\frac{20\!\cdots\!27}{33\!\cdots\!81}a^{6}+\frac{16\!\cdots\!22}{33\!\cdots\!81}a^{5}+\frac{79\!\cdots\!84}{33\!\cdots\!81}a^{4}+\frac{68\!\cdots\!86}{25\!\cdots\!37}a^{3}-\frac{12\!\cdots\!57}{33\!\cdots\!81}a^{2}-\frac{12\!\cdots\!01}{33\!\cdots\!81}a-\frac{24\!\cdots\!39}{25\!\cdots\!37}$, $\frac{6443722864992}{33\!\cdots\!81}a^{12}+\frac{151932223335695}{33\!\cdots\!81}a^{11}-\frac{11\!\cdots\!77}{33\!\cdots\!81}a^{10}-\frac{10\!\cdots\!03}{33\!\cdots\!81}a^{9}+\frac{59\!\cdots\!54}{33\!\cdots\!81}a^{8}+\frac{20\!\cdots\!95}{33\!\cdots\!81}a^{7}-\frac{10\!\cdots\!26}{33\!\cdots\!81}a^{6}-\frac{15\!\cdots\!73}{33\!\cdots\!81}a^{5}+\frac{66\!\cdots\!86}{33\!\cdots\!81}a^{4}+\frac{45\!\cdots\!31}{25\!\cdots\!37}a^{3}-\frac{11\!\cdots\!67}{33\!\cdots\!81}a^{2}-\frac{14\!\cdots\!81}{33\!\cdots\!81}a-\frac{29\!\cdots\!13}{25\!\cdots\!37}$, $\frac{39726459070171}{33\!\cdots\!81}a^{12}+\frac{112751183661847}{33\!\cdots\!81}a^{11}-\frac{18\!\cdots\!07}{33\!\cdots\!81}a^{10}-\frac{12\!\cdots\!50}{33\!\cdots\!81}a^{9}+\frac{21\!\cdots\!66}{33\!\cdots\!81}a^{8}-\frac{46\!\cdots\!46}{33\!\cdots\!81}a^{7}-\frac{46\!\cdots\!39}{33\!\cdots\!81}a^{6}+\frac{84\!\cdots\!48}{33\!\cdots\!81}a^{5}-\frac{90\!\cdots\!55}{33\!\cdots\!81}a^{4}-\frac{35\!\cdots\!43}{25\!\cdots\!37}a^{3}+\frac{24\!\cdots\!90}{33\!\cdots\!81}a^{2}+\frac{80\!\cdots\!89}{33\!\cdots\!81}a+\frac{22\!\cdots\!05}{25\!\cdots\!37}$, $\frac{375233423280744}{33\!\cdots\!81}a^{12}-\frac{277211302117787}{33\!\cdots\!81}a^{11}-\frac{24\!\cdots\!77}{33\!\cdots\!81}a^{10}+\frac{46\!\cdots\!57}{33\!\cdots\!81}a^{9}+\frac{50\!\cdots\!72}{33\!\cdots\!81}a^{8}-\frac{12\!\cdots\!70}{33\!\cdots\!81}a^{7}-\frac{40\!\cdots\!20}{33\!\cdots\!81}a^{6}+\frac{11\!\cdots\!72}{33\!\cdots\!81}a^{5}+\frac{14\!\cdots\!31}{33\!\cdots\!81}a^{4}-\frac{30\!\cdots\!84}{25\!\cdots\!37}a^{3}-\frac{22\!\cdots\!59}{33\!\cdots\!81}a^{2}+\frac{46\!\cdots\!44}{33\!\cdots\!81}a+\frac{17\!\cdots\!53}{25\!\cdots\!37}$, $\frac{308450168928646}{33\!\cdots\!81}a^{12}+\frac{326569043481087}{33\!\cdots\!81}a^{11}-\frac{21\!\cdots\!26}{33\!\cdots\!81}a^{10}-\frac{42\!\cdots\!20}{33\!\cdots\!81}a^{9}+\frac{49\!\cdots\!47}{33\!\cdots\!81}a^{8}-\frac{92\!\cdots\!65}{33\!\cdots\!81}a^{7}-\frac{49\!\cdots\!95}{33\!\cdots\!81}a^{6}-\frac{59\!\cdots\!49}{33\!\cdots\!81}a^{5}+\frac{21\!\cdots\!87}{33\!\cdots\!81}a^{4}+\frac{74\!\cdots\!24}{25\!\cdots\!37}a^{3}-\frac{31\!\cdots\!03}{33\!\cdots\!81}a^{2}-\frac{28\!\cdots\!89}{33\!\cdots\!81}a-\frac{48\!\cdots\!33}{25\!\cdots\!37}$, $\frac{25652125299812}{33\!\cdots\!81}a^{12}-\frac{738771518794870}{33\!\cdots\!81}a^{11}-\frac{22\!\cdots\!00}{33\!\cdots\!81}a^{10}+\frac{46\!\cdots\!38}{33\!\cdots\!81}a^{9}+\frac{13\!\cdots\!14}{33\!\cdots\!81}a^{8}-\frac{87\!\cdots\!67}{33\!\cdots\!81}a^{7}+\frac{48\!\cdots\!00}{33\!\cdots\!81}a^{6}+\frac{61\!\cdots\!78}{33\!\cdots\!81}a^{5}-\frac{37\!\cdots\!38}{33\!\cdots\!81}a^{4}-\frac{13\!\cdots\!30}{25\!\cdots\!37}a^{3}+\frac{29\!\cdots\!46}{33\!\cdots\!81}a^{2}+\frac{19\!\cdots\!87}{33\!\cdots\!81}a+\frac{62\!\cdots\!30}{25\!\cdots\!37}$ (assuming GRH) | 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: | \( 1334709178.87 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 1334709178.87 \cdot 1}{2\cdot\sqrt{224282727500720205065439601}}\cr\approx \mathstrut & 0.365046913382 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 13 |
The 13 conjugacy class representatives for $C_{13}$ |
Character table for $C_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.1.0.1}{1} }^{13}$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(157\) | 157.13.12.1 | $x^{13} + 157$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |