Normalized defining polynomial
\( x^{14} - 5 x^{13} + 11 x^{12} - 11 x^{11} + x^{10} + 5 x^{9} + 13 x^{8} - 41 x^{7} + 38 x^{6} - 6 x^{5} + \cdots + 4 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-145206582282252288\) \(\medspace = -\,2^{12}\cdot 3^{16}\cdot 7^{7}\) | 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: | \(16.82\) | 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: | $2^{6/7}3^{4/3}7^{1/2}\approx 20.7365452398482$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{10}-\frac{1}{6}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{6}+\frac{1}{6}a^{4}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{798}a^{13}-\frac{1}{21}a^{12}-\frac{65}{798}a^{11}-\frac{127}{798}a^{10}-\frac{197}{798}a^{9}+\frac{85}{266}a^{8}-\frac{23}{798}a^{7}+\frac{319}{798}a^{6}-\frac{124}{399}a^{5}+\frac{331}{798}a^{4}-\frac{25}{133}a^{3}-\frac{197}{399}a^{2}-\frac{1}{3}a-\frac{8}{399}$
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)
| |
Fundamental units: | $\frac{23}{114}a^{13}-\frac{1}{2}a^{12}+\frac{22}{57}a^{11}+\frac{31}{57}a^{10}-\frac{11}{19}a^{9}-\frac{20}{19}a^{8}+\frac{163}{57}a^{7}-\frac{9}{19}a^{6}-\frac{289}{114}a^{5}+\frac{17}{38}a^{4}+\frac{71}{19}a^{3}-\frac{28}{57}a^{2}-\frac{1}{3}a+\frac{21}{19}$, $\frac{83}{266}a^{13}-\frac{43}{42}a^{12}+\frac{619}{399}a^{11}-\frac{184}{399}a^{10}-\frac{129}{133}a^{9}+\frac{9}{133}a^{8}+\frac{575}{133}a^{7}-\frac{2246}{399}a^{6}+\frac{1157}{798}a^{5}+\frac{1555}{798}a^{4}+\frac{743}{399}a^{3}-\frac{1705}{399}a^{2}+\frac{10}{3}a+\frac{1}{133}$, $\frac{73}{266}a^{13}-\frac{53}{42}a^{12}+\frac{1991}{798}a^{11}-\frac{493}{266}a^{10}-\frac{583}{798}a^{9}+\frac{261}{266}a^{8}+\frac{1247}{266}a^{7}-\frac{7811}{798}a^{6}+\frac{2104}{399}a^{5}+\frac{311}{133}a^{4}+\frac{599}{399}a^{3}-\frac{3376}{399}a^{2}+6a-\frac{289}{399}$, $\frac{254}{399}a^{13}-\frac{53}{21}a^{12}+\frac{1711}{399}a^{11}-\frac{1003}{399}a^{10}-\frac{562}{399}a^{9}+\frac{44}{133}a^{8}+\frac{3734}{399}a^{7}-\frac{6089}{399}a^{6}+\frac{2710}{399}a^{5}+\frac{139}{133}a^{4}+\frac{2332}{399}a^{3}-\frac{4183}{399}a^{2}+\frac{22}{3}a-\frac{1271}{399}$, $\frac{43}{133}a^{13}-\frac{61}{42}a^{12}+\frac{661}{266}a^{11}-\frac{979}{798}a^{10}-\frac{1483}{798}a^{9}+\frac{251}{266}a^{8}+\frac{1613}{266}a^{7}-\frac{8803}{798}a^{6}+\frac{883}{266}a^{5}+\frac{1735}{399}a^{4}+\frac{200}{133}a^{3}-\frac{1248}{133}a^{2}+\frac{17}{3}a-\frac{601}{399}$, $\frac{709}{798}a^{13}-\frac{137}{42}a^{12}+\frac{676}{133}a^{11}-\frac{799}{399}a^{10}-\frac{1142}{399}a^{9}+\frac{8}{133}a^{8}+\frac{5213}{399}a^{7}-\frac{7213}{399}a^{6}+\frac{1195}{266}a^{5}+\frac{1175}{266}a^{4}+\frac{2951}{399}a^{3}-\frac{1648}{133}a^{2}+\frac{22}{3}a-\frac{751}{399}$ | 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: | \( 2457.12085552 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 2457.12085552 \cdot 1}{2\cdot\sqrt{145206582282252288}}\cr\approx \mathstrut & 1.24641437523 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.144027072.1 |
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: | 7.1.144027072.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.144027072.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |