Normalized defining polynomial
\( x^{13} - x^{12} - x^{11} + x^{10} + 3x^{8} - x^{7} - 5x^{6} + x^{5} + 4x^{4} + 2x^{3} - 3x^{2} - x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1325925503633\) \(\medspace = 97\cdot 13669335089\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(8.56\) | 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: | $97^{1/2}13669335089^{1/2}\approx 1151488.386234529$ | ||
Ramified primes: | \(97\), \(13669335089\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{1325925503633}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{97}a^{12}-\frac{40}{97}a^{11}+\frac{7}{97}a^{10}+\frac{19}{97}a^{9}+\frac{35}{97}a^{8}-\frac{4}{97}a^{7}-\frac{39}{97}a^{6}-\frac{36}{97}a^{5}+\frac{47}{97}a^{4}+\frac{14}{97}a^{3}+\frac{38}{97}a^{2}-\frac{30}{97}a+\frac{5}{97}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a$, $\frac{43}{97}a^{12}+\frac{26}{97}a^{11}-\frac{87}{97}a^{10}-\frac{56}{97}a^{9}+\frac{50}{97}a^{8}+\frac{119}{97}a^{7}+\frac{166}{97}a^{6}-\frac{190}{97}a^{5}-\frac{307}{97}a^{4}+\frac{117}{97}a^{3}+\frac{373}{97}a^{2}+\frac{68}{97}a-\frac{173}{97}$, $\frac{43}{97}a^{12}+\frac{26}{97}a^{11}-\frac{87}{97}a^{10}-\frac{56}{97}a^{9}+\frac{50}{97}a^{8}+\frac{119}{97}a^{7}+\frac{166}{97}a^{6}-\frac{190}{97}a^{5}-\frac{307}{97}a^{4}+\frac{117}{97}a^{3}+\frac{276}{97}a^{2}+\frac{68}{97}a-\frac{173}{97}$, $\frac{161}{97}a^{12}-\frac{135}{97}a^{11}-\frac{134}{97}a^{10}+\frac{52}{97}a^{9}+\frac{9}{97}a^{8}+\frac{520}{97}a^{7}-\frac{71}{97}a^{6}-\frac{655}{97}a^{5}-\frac{96}{97}a^{4}+\frac{411}{97}a^{3}+\frac{492}{97}a^{2}-\frac{174}{97}a-\frac{165}{97}$, $\frac{51}{97}a^{12}-\frac{3}{97}a^{11}-\frac{128}{97}a^{10}-\frac{1}{97}a^{9}+\frac{39}{97}a^{8}+\frac{184}{97}a^{7}+\frac{145}{97}a^{6}-\frac{381}{97}a^{5}-\frac{222}{97}a^{4}+\frac{229}{97}a^{3}+\frac{386}{97}a^{2}+\frac{22}{97}a-\frac{230}{97}$, $\frac{65}{97}a^{12}+\frac{19}{97}a^{11}-\frac{127}{97}a^{10}-\frac{26}{97}a^{9}+\frac{44}{97}a^{8}+\frac{225}{97}a^{7}+\frac{181}{97}a^{6}-\frac{303}{97}a^{5}-\frac{340}{97}a^{4}+\frac{231}{97}a^{3}+\frac{433}{97}a^{2}+\frac{87}{97}a-\frac{160}{97}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 3.1288056301 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 3.1288056301 \cdot 1}{2\cdot\sqrt{1325925503633}}\cr\approx \mathstrut & 0.16718535530 \end{aligned}\]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.3.0.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
97.7.0.1 | $x^{7} + 5 x + 92$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(13669335089\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |