Normalized defining polynomial
\( x^{13} - 4x^{12} + 6x^{11} - 6x^{10} + 7x^{9} - 5x^{8} - x^{7} + x^{6} + 5x^{5} - 2x^{4} - 3x^{3} - x^{2} + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-7805208559959\) \(\medspace = -\,3^{3}\cdot 31\cdot 71\cdot 821\cdot 159977\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.81\) | 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))
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Galois root discriminant: | $3^{1/2}31^{1/2}71^{1/2}821^{1/2}159977^{1/2}\approx 931260.111650338$ | ||
Ramified primes: | \(3\), \(31\), \(71\), \(821\), \(159977\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-867245395551}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{137}a^{12}-\frac{32}{137}a^{11}-\frac{57}{137}a^{10}-\frac{54}{137}a^{9}+\frac{12}{137}a^{8}-\frac{67}{137}a^{7}-\frac{43}{137}a^{6}-\frac{28}{137}a^{5}-\frac{33}{137}a^{4}-\frac{37}{137}a^{3}-\frac{63}{137}a^{2}-\frac{18}{137}a-\frac{44}{137}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{54}{137}a^{12}-\frac{84}{137}a^{11}-\frac{64}{137}a^{10}+\frac{98}{137}a^{9}-\frac{37}{137}a^{8}+\frac{218}{137}a^{7}-\frac{267}{137}a^{6}-\frac{142}{137}a^{5}+\frac{136}{137}a^{4}+\frac{468}{137}a^{3}-\frac{114}{137}a^{2}-\frac{150}{137}a-\frac{47}{137}$, $a$, $\frac{38}{137}a^{12}-\frac{120}{137}a^{11}+\frac{163}{137}a^{10}-\frac{134}{137}a^{9}+\frac{45}{137}a^{8}+\frac{57}{137}a^{7}-\frac{264}{137}a^{6}+\frac{306}{137}a^{5}-\frac{21}{137}a^{4}-\frac{36}{137}a^{3}-\frac{65}{137}a^{2}+\frac{138}{137}a+\frac{109}{137}$, $\frac{19}{137}a^{12}-\frac{60}{137}a^{11}+\frac{13}{137}a^{10}+\frac{70}{137}a^{9}-\frac{46}{137}a^{8}+\frac{97}{137}a^{7}-\frac{132}{137}a^{6}-\frac{121}{137}a^{5}+\frac{332}{137}a^{4}-\frac{18}{137}a^{3}-\frac{101}{137}a^{2}-\frac{205}{137}a-\frac{14}{137}$, $\frac{44}{137}a^{12}-\frac{175}{137}a^{11}+\frac{232}{137}a^{10}-\frac{184}{137}a^{9}+\frac{254}{137}a^{8}-\frac{208}{137}a^{7}+\frac{26}{137}a^{6}-\frac{136}{137}a^{5}+\frac{466}{137}a^{4}-\frac{258}{137}a^{3}-\frac{169}{137}a^{2}-\frac{107}{137}a-\frac{18}{137}$, $\frac{98}{137}a^{12}-\frac{259}{137}a^{11}+\frac{168}{137}a^{10}-\frac{86}{137}a^{9}+\frac{217}{137}a^{8}+\frac{10}{137}a^{7}-\frac{241}{137}a^{6}-\frac{278}{137}a^{5}+\frac{602}{137}a^{4}+\frac{210}{137}a^{3}-\frac{283}{137}a^{2}-\frac{257}{137}a-\frac{65}{137}$, $\frac{85}{137}a^{12}-\frac{254}{137}a^{11}+\frac{361}{137}a^{10}-\frac{480}{137}a^{9}+\frac{472}{137}a^{8}-\frac{215}{137}a^{7}-\frac{93}{137}a^{6}+\frac{86}{137}a^{5}+\frac{72}{137}a^{4}+\frac{6}{137}a^{3}+\frac{262}{137}a^{2}-\frac{23}{137}a-\frac{41}{137}$ | 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: | \( 11.7783503996 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 11.7783503996 \cdot 1}{2\cdot\sqrt{7805208559959}}\cr\approx \mathstrut & 0.165139721269 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ are not computed |
Character table for $S_{13}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 26 sibling: | data not computed |
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 | ${\href{/padicField/2.13.0.1}{13} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.13.0.1}{13} }$ | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.7.0.1 | $x^{7} + 2 x^{2} + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.10.0.1 | $x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.4.0.1 | $x^{4} + 4 x^{2} + 41 x + 7$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
71.4.0.1 | $x^{4} + 4 x^{2} + 41 x + 7$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(821\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(159977\) | $\Q_{159977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{159977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |