Normalized defining polynomial
\( x^{14} - 4 x^{13} + 9 x^{12} - 20 x^{11} + 64 x^{10} - 163 x^{9} + 248 x^{8} - 197 x^{7} + 73 x^{6} - 76 x^{5} + 182 x^{4} - 203 x^{3} + 113 x^{2} - 30 x + 4 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1475921147386413091\) \(\medspace = -\,7^{7}\cdot 13^{11}\) | 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))
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Galois root discriminant: | $7^{1/2}13^{5/6}\approx 22.430303342683025$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-91}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{6}a^{12}+\frac{1}{6}a^{8}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{3}+\frac{1}{6}a$, $\frac{1}{327209214}a^{13}+\frac{3744792}{54534869}a^{12}+\frac{9002329}{109069738}a^{11}+\frac{13953151}{109069738}a^{10}+\frac{161842753}{327209214}a^{9}+\frac{28234525}{109069738}a^{8}-\frac{130631041}{327209214}a^{7}-\frac{55525975}{327209214}a^{6}-\frac{15990289}{54534869}a^{5}-\frac{4320010}{54534869}a^{4}-\frac{19470599}{109069738}a^{3}+\frac{80181827}{163604607}a^{2}+\frac{10801123}{54534869}a-\frac{4718677}{54534869}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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)
| |
Fundamental units: | $\frac{20510785}{109069738}a^{13}-\frac{38566195}{54534869}a^{12}+\frac{229665539}{163604607}a^{11}-\frac{341958799}{109069738}a^{10}+\frac{588382944}{54534869}a^{9}-\frac{1460003892}{54534869}a^{8}+\frac{5752623257}{163604607}a^{7}-\frac{1008317396}{54534869}a^{6}+\frac{291669431}{327209214}a^{5}-\frac{4771160161}{327209214}a^{4}+\frac{3309423761}{109069738}a^{3}-\frac{1160049044}{54534869}a^{2}+\frac{846036321}{109069738}a-\frac{201639763}{163604607}$, $\frac{56261021}{163604607}a^{13}-\frac{133032827}{109069738}a^{12}+\frac{281665689}{109069738}a^{11}-\frac{621591617}{109069738}a^{10}+\frac{3181256117}{163604607}a^{9}-\frac{2584813046}{54534869}a^{8}+\frac{10578305746}{163604607}a^{7}-\frac{6257738648}{163604607}a^{6}+\frac{291422691}{54534869}a^{5}-\frac{2364936159}{109069738}a^{4}+\frac{3025893888}{54534869}a^{3}-\frac{14614882481}{327209214}a^{2}+\frac{1686800843}{109069738}a-\frac{142227949}{54534869}$, $\frac{20510785}{109069738}a^{13}-\frac{38566195}{54534869}a^{12}+\frac{229665539}{163604607}a^{11}-\frac{341958799}{109069738}a^{10}+\frac{588382944}{54534869}a^{9}-\frac{1460003892}{54534869}a^{8}+\frac{5752623257}{163604607}a^{7}-\frac{1008317396}{54534869}a^{6}+\frac{291669431}{327209214}a^{5}-\frac{4771160161}{327209214}a^{4}+\frac{3309423761}{109069738}a^{3}-\frac{1160049044}{54534869}a^{2}+\frac{736966583}{109069738}a-\frac{201639763}{163604607}$, $\frac{5249591}{54534869}a^{13}-\frac{19298729}{54534869}a^{12}+\frac{223231835}{327209214}a^{11}-\frac{83879375}{54534869}a^{10}+\frac{583505455}{109069738}a^{9}-\frac{1436825321}{109069738}a^{8}+\frac{5443033493}{327209214}a^{7}-\frac{821944437}{109069738}a^{6}-\frac{318154337}{327209214}a^{5}-\frac{2226157583}{327209214}a^{4}+\frac{835727649}{54534869}a^{3}-\frac{543688031}{54534869}a^{2}+\frac{192036649}{109069738}a+\frac{14366476}{163604607}$, $\frac{7934144}{54534869}a^{13}-\frac{114921139}{327209214}a^{12}+\frac{69108285}{109069738}a^{11}-\frac{175320545}{109069738}a^{10}+\frac{337767270}{54534869}a^{9}-\frac{2041207958}{163604607}a^{8}+\frac{602154896}{54534869}a^{7}-\frac{39537772}{163604607}a^{6}-\frac{105354337}{163604607}a^{5}-\frac{1014013843}{109069738}a^{4}+\frac{615039032}{54534869}a^{3}-\frac{521690001}{109069738}a^{2}+\frac{42836879}{327209214}a-\frac{15620740}{54534869}$, $\frac{37667629}{327209214}a^{13}-\frac{61579250}{163604607}a^{12}+\frac{123395975}{163604607}a^{11}-\frac{193342495}{109069738}a^{10}+\frac{1000211906}{163604607}a^{9}-\frac{2352138098}{163604607}a^{8}+\frac{989399391}{54534869}a^{7}-\frac{564367906}{54534869}a^{6}+\frac{335307225}{109069738}a^{5}-\frac{2954381329}{327209214}a^{4}+\frac{1649694895}{109069738}a^{3}-\frac{2146232569}{163604607}a^{2}+\frac{1653210545}{327209214}a-\frac{171245269}{163604607}$ | 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: | \( 1466.80774384 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 1466.80774384 \cdot 2}{2\cdot\sqrt{1475921147386413091}}\cr\approx \mathstrut & 0.466767591807 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-91}) \), 7.1.127353499.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 7 sibling: | 7.1.127353499.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.127353499.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(13\) | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |