Normalized defining polynomial
\( x^{13} - 3 x^{12} - 8 x^{11} + 27 x^{10} + 22 x^{9} - 89 x^{8} - 23 x^{7} + 132 x^{6} + 5 x^{5} + \cdots - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[13, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(74057741281094693\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.85\) | 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: | $74057741281094693^{1/2}\approx 272135520.06508577$ | ||
Ramified primes: | \(74057741281094693\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{74057\!\cdots\!94693}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a$, $a^{12}-2a^{11}-10a^{10}+17a^{9}+39a^{8}-50a^{7}-73a^{6}+59a^{5}+65a^{4}-24a^{3}-23a^{2}+3a+2$, $a^{11}-2a^{10}-9a^{9}+15a^{8}+31a^{7}-37a^{6}-50a^{5}+35a^{4}+37a^{3}-13a^{2}-8a+3$, $a^{11}-2a^{10}-9a^{9}+15a^{8}+31a^{7}-37a^{6}-49a^{5}+33a^{4}+33a^{3}-6a^{2}-5a$, $a^{11}-3a^{10}-7a^{9}+24a^{8}+15a^{7}-65a^{6}-8a^{5}+67a^{4}-3a^{3}-20a^{2}+a+1$, $a^{12}-3a^{11}-8a^{10}+27a^{9}+22a^{8}-89a^{7}-23a^{6}+131a^{5}+6a^{4}-82a^{3}+a^{2}+15a-1$, $a^{11}-2a^{10}-9a^{9}+15a^{8}+30a^{7}-35a^{6}-43a^{5}+23a^{4}+22a^{3}+6a^{2}-2$, $a-1$, $a^{11}-3a^{10}-7a^{9}+24a^{8}+15a^{7}-65a^{6}-8a^{5}+67a^{4}-3a^{3}-20a^{2}+a$, $a^{12}-a^{11}-11a^{10}+6a^{9}+46a^{8}-6a^{7}-87a^{6}-15a^{5}+72a^{4}+24a^{3}-20a^{2}-5a+1$, $2a^{12}-5a^{11}-17a^{10}+42a^{9}+52a^{8}-123a^{7}-68a^{6}+149a^{5}+37a^{4}-69a^{3}-9a^{2}+10a+1$, $a^{10}-2a^{9}-9a^{8}+15a^{7}+31a^{6}-37a^{5}-49a^{4}+33a^{3}+33a^{2}-6a-6$ | 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: | \( 10098.9038604 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 10098.9038604 \cdot 1}{2\cdot\sqrt{74057741281094693}}\cr\approx \mathstrut & 0.152001878337 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(74057741281094693\) | $\Q_{74057741281094693}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |