Normalized defining polynomial
\( x^{13} - x^{12} - 312 x^{11} + 765 x^{10} + 31073 x^{9} - 114643 x^{8} - 1071164 x^{7} + 4472586 x^{6} + \cdots - 1052321 \)
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: | \(9269664678331989431355838883693521\) \(\medspace = 677^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(410.06\) | 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: | $677^{12/13}\approx 410.06346656832307$ | ||
Ramified primes: | \(677\) | 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: | \(677\) | ||
Dirichlet character group: | $\lbrace$$\chi_{677}(1,·)$, $\chi_{677}(263,·)$, $\chi_{677}(40,·)$, $\chi_{677}(457,·)$, $\chi_{677}(362,·)$, $\chi_{677}(333,·)$, $\chi_{677}(365,·)$, $\chi_{677}(115,·)$, $\chi_{677}(533,·)$, $\chi_{677}(246,·)$, $\chi_{677}(538,·)$, $\chi_{677}(426,·)$, $\chi_{677}(383,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{251}a^{11}-\frac{120}{251}a^{10}-\frac{104}{251}a^{9}+\frac{1}{251}a^{8}-\frac{12}{251}a^{7}-\frac{30}{251}a^{6}+\frac{101}{251}a^{5}+\frac{24}{251}a^{4}-\frac{109}{251}a^{3}+\frac{72}{251}a^{2}-\frac{99}{251}a+\frac{39}{251}$, $\frac{1}{24\!\cdots\!29}a^{12}-\frac{17\!\cdots\!84}{24\!\cdots\!29}a^{11}-\frac{24\!\cdots\!16}{24\!\cdots\!29}a^{10}+\frac{48\!\cdots\!44}{24\!\cdots\!29}a^{9}-\frac{21\!\cdots\!53}{24\!\cdots\!29}a^{8}-\frac{42\!\cdots\!22}{24\!\cdots\!29}a^{7}-\frac{82\!\cdots\!22}{24\!\cdots\!29}a^{6}-\frac{12\!\cdots\!11}{24\!\cdots\!29}a^{5}-\frac{90\!\cdots\!23}{24\!\cdots\!29}a^{4}-\frac{10\!\cdots\!26}{24\!\cdots\!29}a^{3}-\frac{82\!\cdots\!60}{24\!\cdots\!29}a^{2}+\frac{23\!\cdots\!36}{24\!\cdots\!29}a+\frac{11\!\cdots\!66}{24\!\cdots\!29}$
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: | $\frac{35\!\cdots\!57}{24\!\cdots\!29}a^{12}+\frac{88\!\cdots\!72}{24\!\cdots\!29}a^{11}-\frac{10\!\cdots\!90}{24\!\cdots\!29}a^{10}-\frac{63\!\cdots\!77}{24\!\cdots\!29}a^{9}+\frac{96\!\cdots\!84}{24\!\cdots\!29}a^{8}-\frac{11\!\cdots\!19}{24\!\cdots\!29}a^{7}-\frac{32\!\cdots\!50}{24\!\cdots\!29}a^{6}+\frac{59\!\cdots\!79}{24\!\cdots\!29}a^{5}+\frac{43\!\cdots\!88}{24\!\cdots\!29}a^{4}-\frac{85\!\cdots\!40}{24\!\cdots\!29}a^{3}-\frac{19\!\cdots\!01}{24\!\cdots\!29}a^{2}+\frac{32\!\cdots\!59}{24\!\cdots\!29}a+\frac{73\!\cdots\!61}{24\!\cdots\!29}$, $\frac{53\!\cdots\!13}{24\!\cdots\!29}a^{12}+\frac{27\!\cdots\!64}{24\!\cdots\!29}a^{11}-\frac{16\!\cdots\!57}{24\!\cdots\!29}a^{10}+\frac{15\!\cdots\!14}{24\!\cdots\!29}a^{9}+\frac{16\!\cdots\!56}{24\!\cdots\!29}a^{8}-\frac{35\!\cdots\!93}{24\!\cdots\!29}a^{7}-\frac{62\!\cdots\!64}{24\!\cdots\!29}a^{6}+\frac{14\!\cdots\!00}{24\!\cdots\!29}a^{5}+\frac{95\!\cdots\!84}{24\!\cdots\!29}a^{4}-\frac{18\!\cdots\!53}{24\!\cdots\!29}a^{3}-\frac{51\!\cdots\!39}{24\!\cdots\!29}a^{2}+\frac{49\!\cdots\!60}{24\!\cdots\!29}a+\frac{35\!\cdots\!42}{24\!\cdots\!29}$, $\frac{17\!\cdots\!97}{24\!\cdots\!29}a^{12}+\frac{67\!\cdots\!53}{24\!\cdots\!29}a^{11}-\frac{52\!\cdots\!07}{24\!\cdots\!29}a^{10}-\frac{11\!\cdots\!19}{24\!\cdots\!29}a^{9}+\frac{50\!\cdots\!55}{24\!\cdots\!29}a^{8}+\frac{33\!\cdots\!72}{24\!\cdots\!29}a^{7}-\frac{17\!\cdots\!34}{24\!\cdots\!29}a^{6}-\frac{37\!\cdots\!30}{24\!\cdots\!29}a^{5}+\frac{23\!\cdots\!46}{24\!\cdots\!29}a^{4}+\frac{44\!\cdots\!18}{24\!\cdots\!29}a^{3}-\frac{77\!\cdots\!41}{24\!\cdots\!29}a^{2}+\frac{51\!\cdots\!37}{24\!\cdots\!29}a+\frac{38\!\cdots\!53}{24\!\cdots\!29}$, $\frac{48\!\cdots\!59}{24\!\cdots\!29}a^{12}-\frac{18\!\cdots\!29}{24\!\cdots\!29}a^{11}-\frac{14\!\cdots\!38}{24\!\cdots\!29}a^{10}+\frac{77\!\cdots\!50}{24\!\cdots\!29}a^{9}+\frac{12\!\cdots\!96}{24\!\cdots\!29}a^{8}-\frac{91\!\cdots\!11}{24\!\cdots\!29}a^{7}-\frac{25\!\cdots\!95}{24\!\cdots\!29}a^{6}+\frac{28\!\cdots\!02}{24\!\cdots\!29}a^{5}-\frac{17\!\cdots\!81}{24\!\cdots\!29}a^{4}-\frac{25\!\cdots\!00}{24\!\cdots\!29}a^{3}+\frac{54\!\cdots\!69}{24\!\cdots\!29}a^{2}-\frac{35\!\cdots\!73}{24\!\cdots\!29}a+\frac{55\!\cdots\!59}{24\!\cdots\!29}$, $\frac{48\!\cdots\!31}{24\!\cdots\!29}a^{12}-\frac{50\!\cdots\!68}{24\!\cdots\!29}a^{11}-\frac{13\!\cdots\!78}{24\!\cdots\!29}a^{10}+\frac{16\!\cdots\!19}{24\!\cdots\!29}a^{9}+\frac{83\!\cdots\!50}{24\!\cdots\!29}a^{8}-\frac{16\!\cdots\!45}{24\!\cdots\!29}a^{7}+\frac{21\!\cdots\!60}{24\!\cdots\!29}a^{6}+\frac{35\!\cdots\!99}{24\!\cdots\!29}a^{5}-\frac{10\!\cdots\!01}{24\!\cdots\!29}a^{4}-\frac{14\!\cdots\!71}{24\!\cdots\!29}a^{3}+\frac{61\!\cdots\!10}{24\!\cdots\!29}a^{2}-\frac{36\!\cdots\!64}{24\!\cdots\!29}a-\frac{41\!\cdots\!10}{24\!\cdots\!29}$, $\frac{84\!\cdots\!90}{24\!\cdots\!29}a^{12}-\frac{15\!\cdots\!11}{24\!\cdots\!29}a^{11}-\frac{26\!\cdots\!53}{24\!\cdots\!29}a^{10}+\frac{39\!\cdots\!53}{24\!\cdots\!29}a^{9}+\frac{26\!\cdots\!50}{24\!\cdots\!29}a^{8}-\frac{71\!\cdots\!51}{24\!\cdots\!29}a^{7}-\frac{96\!\cdots\!64}{24\!\cdots\!29}a^{6}+\frac{28\!\cdots\!47}{24\!\cdots\!29}a^{5}+\frac{14\!\cdots\!01}{24\!\cdots\!29}a^{4}-\frac{38\!\cdots\!99}{24\!\cdots\!29}a^{3}-\frac{70\!\cdots\!69}{24\!\cdots\!29}a^{2}+\frac{13\!\cdots\!97}{24\!\cdots\!29}a+\frac{10\!\cdots\!09}{24\!\cdots\!29}$, $\frac{88\!\cdots\!88}{24\!\cdots\!29}a^{12}-\frac{19\!\cdots\!16}{24\!\cdots\!29}a^{11}-\frac{25\!\cdots\!33}{24\!\cdots\!29}a^{10}+\frac{95\!\cdots\!77}{24\!\cdots\!29}a^{9}+\frac{20\!\cdots\!67}{24\!\cdots\!29}a^{8}-\frac{11\!\cdots\!97}{24\!\cdots\!29}a^{7}-\frac{32\!\cdots\!29}{24\!\cdots\!29}a^{6}+\frac{24\!\cdots\!63}{24\!\cdots\!29}a^{5}-\frac{10\!\cdots\!78}{24\!\cdots\!29}a^{4}-\frac{97\!\cdots\!42}{24\!\cdots\!29}a^{3}+\frac{21\!\cdots\!11}{24\!\cdots\!29}a^{2}-\frac{11\!\cdots\!10}{24\!\cdots\!29}a-\frac{64\!\cdots\!25}{24\!\cdots\!29}$, $\frac{15\!\cdots\!07}{24\!\cdots\!29}a^{12}+\frac{87\!\cdots\!37}{24\!\cdots\!29}a^{11}-\frac{44\!\cdots\!31}{24\!\cdots\!29}a^{10}-\frac{18\!\cdots\!86}{24\!\cdots\!29}a^{9}+\frac{40\!\cdots\!71}{24\!\cdots\!29}a^{8}+\frac{10\!\cdots\!12}{24\!\cdots\!29}a^{7}-\frac{14\!\cdots\!16}{24\!\cdots\!29}a^{6}-\frac{29\!\cdots\!94}{24\!\cdots\!29}a^{5}+\frac{18\!\cdots\!11}{24\!\cdots\!29}a^{4}+\frac{34\!\cdots\!15}{24\!\cdots\!29}a^{3}-\frac{64\!\cdots\!18}{24\!\cdots\!29}a^{2}-\frac{71\!\cdots\!24}{24\!\cdots\!29}a+\frac{76\!\cdots\!38}{24\!\cdots\!29}$, $\frac{98\!\cdots\!61}{24\!\cdots\!29}a^{12}-\frac{53\!\cdots\!66}{24\!\cdots\!29}a^{11}-\frac{30\!\cdots\!96}{24\!\cdots\!29}a^{10}+\frac{62\!\cdots\!99}{24\!\cdots\!29}a^{9}+\frac{30\!\cdots\!85}{24\!\cdots\!29}a^{8}-\frac{10\!\cdots\!46}{24\!\cdots\!29}a^{7}-\frac{10\!\cdots\!97}{24\!\cdots\!29}a^{6}+\frac{40\!\cdots\!17}{24\!\cdots\!29}a^{5}+\frac{13\!\cdots\!73}{24\!\cdots\!29}a^{4}-\frac{55\!\cdots\!63}{24\!\cdots\!29}a^{3}-\frac{47\!\cdots\!96}{24\!\cdots\!29}a^{2}+\frac{21\!\cdots\!54}{24\!\cdots\!29}a-\frac{11\!\cdots\!73}{24\!\cdots\!29}$, $\frac{41\!\cdots\!13}{24\!\cdots\!29}a^{12}+\frac{15\!\cdots\!69}{24\!\cdots\!29}a^{11}-\frac{12\!\cdots\!01}{24\!\cdots\!29}a^{10}-\frac{27\!\cdots\!49}{24\!\cdots\!29}a^{9}+\frac{11\!\cdots\!46}{24\!\cdots\!29}a^{8}+\frac{87\!\cdots\!08}{24\!\cdots\!29}a^{7}-\frac{41\!\cdots\!81}{24\!\cdots\!29}a^{6}-\frac{11\!\cdots\!84}{24\!\cdots\!29}a^{5}+\frac{54\!\cdots\!39}{24\!\cdots\!29}a^{4}+\frac{26\!\cdots\!54}{24\!\cdots\!29}a^{3}-\frac{19\!\cdots\!00}{24\!\cdots\!29}a^{2}+\frac{12\!\cdots\!75}{24\!\cdots\!29}a+\frac{12\!\cdots\!98}{24\!\cdots\!29}$, $\frac{17\!\cdots\!58}{24\!\cdots\!29}a^{12}+\frac{68\!\cdots\!32}{24\!\cdots\!29}a^{11}-\frac{52\!\cdots\!62}{24\!\cdots\!29}a^{10}-\frac{11\!\cdots\!75}{24\!\cdots\!29}a^{9}+\frac{50\!\cdots\!56}{24\!\cdots\!29}a^{8}+\frac{36\!\cdots\!12}{24\!\cdots\!29}a^{7}-\frac{17\!\cdots\!66}{24\!\cdots\!29}a^{6}-\frac{41\!\cdots\!86}{24\!\cdots\!29}a^{5}+\frac{22\!\cdots\!55}{24\!\cdots\!29}a^{4}+\frac{20\!\cdots\!94}{24\!\cdots\!29}a^{3}-\frac{77\!\cdots\!86}{24\!\cdots\!29}a^{2}+\frac{52\!\cdots\!96}{24\!\cdots\!29}a+\frac{39\!\cdots\!42}{24\!\cdots\!29}$, $\frac{17\!\cdots\!84}{24\!\cdots\!29}a^{12}+\frac{68\!\cdots\!52}{24\!\cdots\!29}a^{11}-\frac{52\!\cdots\!29}{24\!\cdots\!29}a^{10}-\frac{11\!\cdots\!40}{24\!\cdots\!29}a^{9}+\frac{49\!\cdots\!71}{24\!\cdots\!29}a^{8}+\frac{35\!\cdots\!17}{24\!\cdots\!29}a^{7}-\frac{17\!\cdots\!41}{24\!\cdots\!29}a^{6}-\frac{40\!\cdots\!08}{24\!\cdots\!29}a^{5}+\frac{22\!\cdots\!80}{24\!\cdots\!29}a^{4}+\frac{17\!\cdots\!01}{24\!\cdots\!29}a^{3}-\frac{77\!\cdots\!08}{24\!\cdots\!29}a^{2}+\frac{51\!\cdots\!13}{24\!\cdots\!29}a+\frac{38\!\cdots\!91}{24\!\cdots\!29}$ (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: | \( 3454123871256.3633 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 3454123871256.3633 \cdot 1}{2\cdot\sqrt{9269664678331989431355838883693521}}\cr\approx \mathstrut & 0.146948732341942 \end{aligned}\] (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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(677\) | Deg $13$ | $13$ | $1$ | $12$ |