Normalized defining polynomial
\( x^{13} - 2x^{10} + 2x^{9} - x^{8} - 3x^{7} - 3x^{6} + 9x^{5} - 10x^{4} + 7x^{3} + 1 \)
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)
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Discriminant: | \(-9197165262683\) \(\medspace = -\,82163\cdot 111938041\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.94\) | 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: | $82163^{1/2}111938041^{1/2}\approx 3032682.849010592$ | ||
Ramified primes: | \(82163\), \(111938041\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-9197165262683}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7979}a^{12}-\frac{1670}{7979}a^{11}-\frac{3750}{7979}a^{10}-\frac{1017}{7979}a^{9}-\frac{1135}{7979}a^{8}-\frac{3553}{7979}a^{7}-\frac{2869}{7979}a^{6}+\frac{3827}{7979}a^{5}+\frac{98}{7979}a^{4}+\frac{3889}{7979}a^{3}+\frac{283}{7979}a^{2}-\frac{1849}{7979}a-\frac{43}{7979}$
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)
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Fundamental units: | $\frac{2597}{7979}a^{12}-\frac{4393}{7979}a^{11}-\frac{4370}{7979}a^{10}-\frac{8079}{7979}a^{9}+\frac{12614}{7979}a^{8}-\frac{3417}{7979}a^{7}-\frac{6386}{7979}a^{6}+\frac{4864}{7979}a^{5}+\frac{47052}{7979}a^{4}-\frac{49555}{7979}a^{3}+\frac{32799}{7979}a^{2}-\frac{6474}{7979}a-\frac{7944}{7979}$, $\frac{6169}{7979}a^{12}-\frac{1341}{7979}a^{11}-\frac{2629}{7979}a^{10}-\frac{10358}{7979}a^{9}+\frac{19705}{7979}a^{8}-\frac{144}{7979}a^{7}-\frac{25376}{7979}a^{6}-\frac{17056}{7979}a^{5}+\frac{69969}{7979}a^{4}-\frac{65444}{7979}a^{3}+\frac{14384}{7979}a^{2}+\frac{11468}{7979}a-\frac{9939}{7979}$, $a$, $\frac{3882}{7979}a^{12}+\frac{3987}{7979}a^{11}+\frac{4175}{7979}a^{10}-\frac{6368}{7979}a^{9}-\frac{1662}{7979}a^{8}-\frac{5034}{7979}a^{7}-\frac{6753}{7979}a^{6}-\frac{16442}{7979}a^{5}+\frac{13402}{7979}a^{4}-\frac{23107}{7979}a^{3}+\frac{21441}{7979}a^{2}+\frac{3282}{7979}a+\frac{633}{7979}$, $\frac{5929}{7979}a^{12}+\frac{509}{7979}a^{11}-\frac{4256}{7979}a^{10}-\frac{13627}{7979}a^{9}+\frac{12840}{7979}a^{8}+\frac{6802}{7979}a^{7}-\frac{23010}{7979}a^{6}-\frac{25930}{7979}a^{5}+\frac{62407}{7979}a^{4}-\frac{33345}{7979}a^{3}+\frac{2317}{7979}a^{2}+\frac{16383}{7979}a-\frac{7598}{7979}$, $\frac{2809}{7979}a^{12}+\frac{622}{7979}a^{11}-\frac{1470}{7979}a^{10}-\frac{8250}{7979}a^{9}+\frac{3385}{7979}a^{8}+\frac{1352}{7979}a^{7}-\frac{8210}{7979}a^{6}-\frac{13628}{7979}a^{5}+\frac{27933}{7979}a^{4}-\frac{7029}{7979}a^{3}+\frac{13005}{7979}a^{2}+\frac{488}{7979}a+\frac{6877}{7979}$, $\frac{1830}{7979}a^{12}-\frac{143}{7979}a^{11}-\frac{560}{7979}a^{10}-\frac{2003}{7979}a^{9}+\frac{5469}{7979}a^{8}+\frac{895}{7979}a^{7}-\frac{8067}{7979}a^{6}-\frac{2152}{7979}a^{5}+\frac{19760}{7979}a^{4}-\frac{24335}{7979}a^{3}-\frac{745}{7979}a^{2}+\frac{7405}{7979}a-\frac{6879}{7979}$ | 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: | \( 13.0679687985 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 13.0679687985 \cdot 1}{2\cdot\sqrt{9197165262683}}\cr\approx \mathstrut & 0.168787556802 \end{aligned}\]
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 | ${\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(82163\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(111938041\) | $\Q_{111938041}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |