Normalized defining polynomial
\( x^{13} - 65 x^{11} - 403 x^{10} + 130 x^{9} + 2639 x^{8} - 20579 x^{7} - 202852 x^{6} - 634751 x^{5} + \cdots - 3741300 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(395701761602916103810493018169\) \(\medspace = 3^{6}\cdot 13^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(189.11\) | 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: | $3^{1/2}13^{24/13}\approx 197.27471817203326$ | ||
Ramified primes: | \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{36}a^{8}-\frac{1}{18}a^{7}-\frac{1}{18}a^{6}-\frac{2}{9}a^{5}-\frac{1}{18}a^{4}+\frac{5}{18}a^{3}+\frac{5}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{36}a^{9}+\frac{1}{6}a^{5}-\frac{13}{36}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{216}a^{10}-\frac{1}{72}a^{9}+\frac{1}{216}a^{8}-\frac{7}{108}a^{7}+\frac{2}{27}a^{6}-\frac{5}{54}a^{5}+\frac{5}{72}a^{4}-\frac{17}{216}a^{3}+\frac{17}{72}a^{2}-\frac{7}{36}a+\frac{1}{6}$, $\frac{1}{44627760}a^{11}-\frac{26317}{22313880}a^{10}-\frac{272767}{22313880}a^{9}-\frac{367837}{44627760}a^{8}+\frac{115159}{22313880}a^{7}+\frac{451463}{11156940}a^{6}-\frac{6559937}{44627760}a^{5}-\frac{537127}{22313880}a^{4}+\frac{997255}{4462776}a^{3}-\frac{173843}{550960}a^{2}+\frac{2906303}{7437960}a-\frac{84541}{247932}$, $\frac{1}{42\!\cdots\!00}a^{12}-\frac{33\!\cdots\!79}{42\!\cdots\!00}a^{11}+\frac{12\!\cdots\!04}{66\!\cdots\!25}a^{10}-\frac{18\!\cdots\!49}{14\!\cdots\!00}a^{9}-\frac{11\!\cdots\!77}{42\!\cdots\!00}a^{8}+\frac{10\!\cdots\!31}{21\!\cdots\!00}a^{7}+\frac{15\!\cdots\!07}{46\!\cdots\!00}a^{6}+\frac{72\!\cdots\!31}{42\!\cdots\!00}a^{5}-\frac{64\!\cdots\!89}{10\!\cdots\!20}a^{4}+\frac{10\!\cdots\!87}{42\!\cdots\!00}a^{3}-\frac{20\!\cdots\!09}{14\!\cdots\!00}a^{2}-\frac{58\!\cdots\!17}{14\!\cdots\!60}a-\frac{65\!\cdots\!11}{46\!\cdots\!52}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $3$ |
Class group and class number
$C_{13}$, which has order $13$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{96\!\cdots\!01}{14\!\cdots\!00}a^{12}-\frac{19\!\cdots\!99}{14\!\cdots\!00}a^{11}-\frac{11\!\cdots\!26}{22\!\cdots\!75}a^{10}-\frac{11\!\cdots\!89}{46\!\cdots\!00}a^{9}+\frac{78\!\cdots\!63}{14\!\cdots\!00}a^{8}+\frac{20\!\cdots\!51}{70\!\cdots\!00}a^{7}-\frac{10\!\cdots\!59}{46\!\cdots\!00}a^{6}-\frac{20\!\cdots\!29}{14\!\cdots\!00}a^{5}-\frac{69\!\cdots\!77}{35\!\cdots\!40}a^{4}-\frac{26\!\cdots\!13}{14\!\cdots\!00}a^{3}-\frac{20\!\cdots\!89}{46\!\cdots\!00}a^{2}-\frac{33\!\cdots\!29}{46\!\cdots\!20}a-\frac{55\!\cdots\!07}{15\!\cdots\!84}$, $\frac{25\!\cdots\!89}{52\!\cdots\!00}a^{12}-\frac{31\!\cdots\!67}{10\!\cdots\!00}a^{11}-\frac{84\!\cdots\!53}{26\!\cdots\!00}a^{10}-\frac{31\!\cdots\!41}{17\!\cdots\!00}a^{9}+\frac{21\!\cdots\!29}{10\!\cdots\!00}a^{8}+\frac{68\!\cdots\!73}{52\!\cdots\!00}a^{7}-\frac{19\!\cdots\!41}{17\!\cdots\!00}a^{6}-\frac{98\!\cdots\!97}{10\!\cdots\!00}a^{5}-\frac{12\!\cdots\!23}{52\!\cdots\!60}a^{4}-\frac{26\!\cdots\!57}{52\!\cdots\!00}a^{3}-\frac{11\!\cdots\!47}{35\!\cdots\!00}a^{2}-\frac{17\!\cdots\!59}{35\!\cdots\!40}a-\frac{15\!\cdots\!47}{11\!\cdots\!88}$, $\frac{81\!\cdots\!89}{21\!\cdots\!00}a^{12}+\frac{18\!\cdots\!99}{21\!\cdots\!00}a^{11}-\frac{79\!\cdots\!17}{26\!\cdots\!00}a^{10}-\frac{14\!\cdots\!01}{70\!\cdots\!00}a^{9}+\frac{93\!\cdots\!37}{21\!\cdots\!00}a^{8}+\frac{29\!\cdots\!29}{10\!\cdots\!00}a^{7}-\frac{58\!\cdots\!11}{70\!\cdots\!00}a^{6}-\frac{23\!\cdots\!51}{21\!\cdots\!00}a^{5}-\frac{37\!\cdots\!31}{13\!\cdots\!65}a^{4}-\frac{72\!\cdots\!57}{21\!\cdots\!00}a^{3}-\frac{11\!\cdots\!31}{70\!\cdots\!00}a^{2}+\frac{16\!\cdots\!13}{14\!\cdots\!56}a+\frac{59\!\cdots\!03}{23\!\cdots\!76}$, $\frac{27\!\cdots\!41}{70\!\cdots\!00}a^{12}-\frac{46\!\cdots\!49}{70\!\cdots\!00}a^{11}-\frac{23\!\cdots\!83}{88\!\cdots\!00}a^{10}-\frac{24\!\cdots\!29}{23\!\cdots\!00}a^{9}+\frac{26\!\cdots\!13}{70\!\cdots\!00}a^{8}+\frac{34\!\cdots\!81}{35\!\cdots\!00}a^{7}-\frac{29\!\cdots\!79}{23\!\cdots\!00}a^{6}-\frac{42\!\cdots\!59}{70\!\cdots\!00}a^{5}-\frac{71\!\cdots\!49}{88\!\cdots\!10}a^{4}-\frac{44\!\cdots\!33}{70\!\cdots\!00}a^{3}-\frac{43\!\cdots\!59}{23\!\cdots\!00}a^{2}-\frac{62\!\cdots\!43}{23\!\cdots\!60}a-\frac{51\!\cdots\!17}{78\!\cdots\!92}$, $\frac{17\!\cdots\!47}{42\!\cdots\!00}a^{12}+\frac{26\!\cdots\!47}{42\!\cdots\!00}a^{11}-\frac{22\!\cdots\!22}{66\!\cdots\!25}a^{10}-\frac{83\!\cdots\!83}{14\!\cdots\!00}a^{9}-\frac{81\!\cdots\!39}{42\!\cdots\!00}a^{8}+\frac{12\!\cdots\!97}{21\!\cdots\!00}a^{7}+\frac{14\!\cdots\!27}{14\!\cdots\!00}a^{6}-\frac{10\!\cdots\!63}{42\!\cdots\!00}a^{5}-\frac{14\!\cdots\!59}{10\!\cdots\!20}a^{4}-\frac{11\!\cdots\!11}{42\!\cdots\!00}a^{3}-\frac{45\!\cdots\!83}{14\!\cdots\!00}a^{2}+\frac{74\!\cdots\!77}{14\!\cdots\!60}a+\frac{13\!\cdots\!35}{46\!\cdots\!52}$, $\frac{77\!\cdots\!13}{86\!\cdots\!00}a^{12}-\frac{54\!\cdots\!03}{78\!\cdots\!00}a^{11}-\frac{56\!\cdots\!51}{97\!\cdots\!00}a^{10}-\frac{24\!\cdots\!59}{78\!\cdots\!00}a^{9}+\frac{91\!\cdots\!37}{26\!\cdots\!00}a^{8}+\frac{29\!\cdots\!29}{13\!\cdots\!00}a^{7}-\frac{17\!\cdots\!61}{86\!\cdots\!00}a^{6}-\frac{12\!\cdots\!53}{78\!\cdots\!00}a^{5}-\frac{21\!\cdots\!68}{48\!\cdots\!95}a^{4}-\frac{71\!\cdots\!21}{78\!\cdots\!00}a^{3}-\frac{18\!\cdots\!93}{26\!\cdots\!00}a^{2}-\frac{57\!\cdots\!97}{52\!\cdots\!28}a-\frac{26\!\cdots\!39}{86\!\cdots\!88}$ (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: | \( 49220254401.852295 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 49220254401.852295 \cdot 13}{2\cdot\sqrt{395701761602916103810493018169}}\cr\approx \mathstrut & 62.5867535410518 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 26 |
The 8 conjugacy class representatives for $D_{13}$ |
Character table for $D_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | deg 26 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(13\) | 13.13.24.1 | $x^{13} + 156 x^{12} + 13$ | $13$ | $1$ | $24$ | $C_{13}$ | $[2]$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.85683.13t2.a.a | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.85683.13t2.a.c | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.85683.13t2.a.f | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.85683.13t2.a.e | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.85683.13t2.a.d | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.85683.13t2.a.b | $2$ | $ 3 \cdot 13^{4}$ | 13.1.395701761602916103810493018169.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |