Normalized defining polynomial
\( x^{14} + 7 x^{12} + 105 x^{10} - 168 x^{9} + 455 x^{8} - 480 x^{7} + 539 x^{6} - 252 x^{5} + 441 x^{4} + \cdots + 1117 \)
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: | \(-10334408033917410766848\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.36\) | 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: | $2^{6/7}3^{6/7}7^{47/42}\approx 40.99130413528715$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{52\!\cdots\!60}a^{13}+\frac{10\!\cdots\!84}{13\!\cdots\!15}a^{12}+\frac{24\!\cdots\!99}{26\!\cdots\!30}a^{11}+\frac{28\!\cdots\!99}{26\!\cdots\!30}a^{10}+\frac{14\!\cdots\!59}{26\!\cdots\!30}a^{9}-\frac{18\!\cdots\!61}{10\!\cdots\!32}a^{8}+\frac{10\!\cdots\!39}{26\!\cdots\!83}a^{7}+\frac{57\!\cdots\!47}{26\!\cdots\!83}a^{6}-\frac{20\!\cdots\!71}{52\!\cdots\!60}a^{5}+\frac{84\!\cdots\!31}{26\!\cdots\!30}a^{4}-\frac{90\!\cdots\!81}{26\!\cdots\!30}a^{3}-\frac{64\!\cdots\!74}{13\!\cdots\!15}a^{2}-\frac{54\!\cdots\!48}{13\!\cdots\!15}a+\frac{20\!\cdots\!93}{52\!\cdots\!60}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{23\!\cdots\!51}{52\!\cdots\!60}a^{13}+\frac{76\!\cdots\!41}{52\!\cdots\!60}a^{12}+\frac{15\!\cdots\!93}{52\!\cdots\!60}a^{11}+\frac{30\!\cdots\!89}{26\!\cdots\!30}a^{10}+\frac{11\!\cdots\!29}{26\!\cdots\!30}a^{9}-\frac{61\!\cdots\!33}{10\!\cdots\!32}a^{8}+\frac{81\!\cdots\!33}{52\!\cdots\!66}a^{7}-\frac{26\!\cdots\!91}{26\!\cdots\!83}a^{6}+\frac{42\!\cdots\!99}{52\!\cdots\!60}a^{5}+\frac{19\!\cdots\!17}{52\!\cdots\!60}a^{4}+\frac{56\!\cdots\!63}{52\!\cdots\!60}a^{3}-\frac{32\!\cdots\!99}{13\!\cdots\!15}a^{2}+\frac{23\!\cdots\!59}{26\!\cdots\!30}a-\frac{36\!\cdots\!47}{52\!\cdots\!60}$, $\frac{12\!\cdots\!11}{52\!\cdots\!60}a^{13}+\frac{42\!\cdots\!31}{52\!\cdots\!60}a^{12}+\frac{91\!\cdots\!73}{52\!\cdots\!60}a^{11}+\frac{14\!\cdots\!19}{26\!\cdots\!30}a^{10}+\frac{68\!\cdots\!79}{26\!\cdots\!30}a^{9}-\frac{34\!\cdots\!29}{10\!\cdots\!32}a^{8}+\frac{53\!\cdots\!65}{52\!\cdots\!66}a^{7}-\frac{23\!\cdots\!26}{26\!\cdots\!83}a^{6}+\frac{61\!\cdots\!99}{52\!\cdots\!60}a^{5}-\frac{34\!\cdots\!53}{52\!\cdots\!60}a^{4}+\frac{72\!\cdots\!63}{52\!\cdots\!60}a^{3}-\frac{98\!\cdots\!19}{13\!\cdots\!15}a^{2}+\frac{64\!\cdots\!09}{26\!\cdots\!30}a-\frac{45\!\cdots\!47}{52\!\cdots\!60}$, $\frac{43\!\cdots\!67}{52\!\cdots\!60}a^{13}-\frac{27\!\cdots\!79}{26\!\cdots\!30}a^{12}+\frac{25\!\cdots\!11}{52\!\cdots\!60}a^{11}-\frac{58\!\cdots\!69}{52\!\cdots\!60}a^{10}+\frac{99\!\cdots\!99}{13\!\cdots\!15}a^{9}-\frac{22\!\cdots\!87}{10\!\cdots\!32}a^{8}+\frac{21\!\cdots\!15}{52\!\cdots\!66}a^{7}-\frac{17\!\cdots\!25}{26\!\cdots\!83}a^{6}-\frac{55\!\cdots\!47}{52\!\cdots\!60}a^{5}+\frac{79\!\cdots\!97}{26\!\cdots\!30}a^{4}-\frac{28\!\cdots\!29}{52\!\cdots\!60}a^{3}+\frac{34\!\cdots\!73}{52\!\cdots\!60}a^{2}-\frac{60\!\cdots\!81}{13\!\cdots\!15}a+\frac{81\!\cdots\!71}{52\!\cdots\!60}$, $\frac{12\!\cdots\!67}{52\!\cdots\!60}a^{13}+\frac{58\!\cdots\!51}{26\!\cdots\!30}a^{12}+\frac{82\!\cdots\!71}{52\!\cdots\!60}a^{11}+\frac{98\!\cdots\!11}{52\!\cdots\!60}a^{10}+\frac{30\!\cdots\!64}{13\!\cdots\!15}a^{9}-\frac{14\!\cdots\!79}{10\!\cdots\!32}a^{8}+\frac{30\!\cdots\!81}{52\!\cdots\!66}a^{7}+\frac{83\!\cdots\!91}{26\!\cdots\!83}a^{6}-\frac{79\!\cdots\!67}{52\!\cdots\!60}a^{5}+\frac{10\!\cdots\!77}{26\!\cdots\!30}a^{4}-\frac{26\!\cdots\!09}{52\!\cdots\!60}a^{3}+\frac{32\!\cdots\!13}{52\!\cdots\!60}a^{2}-\frac{23\!\cdots\!71}{13\!\cdots\!15}a-\frac{25\!\cdots\!09}{52\!\cdots\!60}$, $\frac{19\!\cdots\!35}{10\!\cdots\!32}a^{13}-\frac{74\!\cdots\!89}{10\!\cdots\!32}a^{12}+\frac{14\!\cdots\!67}{10\!\cdots\!32}a^{11}-\frac{29\!\cdots\!91}{52\!\cdots\!66}a^{10}+\frac{52\!\cdots\!14}{26\!\cdots\!83}a^{9}-\frac{11\!\cdots\!61}{10\!\cdots\!32}a^{8}+\frac{11\!\cdots\!01}{52\!\cdots\!66}a^{7}-\frac{12\!\cdots\!02}{26\!\cdots\!83}a^{6}+\frac{64\!\cdots\!63}{10\!\cdots\!32}a^{5}-\frac{76\!\cdots\!97}{10\!\cdots\!32}a^{4}+\frac{51\!\cdots\!97}{10\!\cdots\!32}a^{3}-\frac{72\!\cdots\!72}{26\!\cdots\!83}a^{2}-\frac{34\!\cdots\!96}{26\!\cdots\!83}a-\frac{25\!\cdots\!59}{10\!\cdots\!32}$, $\frac{30\!\cdots\!65}{10\!\cdots\!32}a^{13}-\frac{30\!\cdots\!17}{10\!\cdots\!32}a^{12}+\frac{25\!\cdots\!69}{10\!\cdots\!32}a^{11}-\frac{10\!\cdots\!99}{52\!\cdots\!66}a^{10}+\frac{86\!\cdots\!85}{26\!\cdots\!83}a^{9}-\frac{82\!\cdots\!59}{10\!\cdots\!32}a^{8}+\frac{11\!\cdots\!95}{52\!\cdots\!66}a^{7}-\frac{88\!\cdots\!25}{26\!\cdots\!83}a^{6}+\frac{49\!\cdots\!37}{10\!\cdots\!32}a^{5}-\frac{32\!\cdots\!57}{10\!\cdots\!32}a^{4}+\frac{35\!\cdots\!63}{10\!\cdots\!32}a^{3}+\frac{13\!\cdots\!76}{26\!\cdots\!83}a^{2}-\frac{16\!\cdots\!19}{26\!\cdots\!83}a+\frac{40\!\cdots\!31}{10\!\cdots\!32}$ (assuming GRH) | 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: | \( 586605.1191605367 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 586605.1191605367 \cdot 1}{2\cdot\sqrt{10334408033917410766848}}\cr\approx \mathstrut & 1.11540382401034 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.38423222208.4 |
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: | 7.1.38423222208.4 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.38423222208.4 |
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 | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.14.15.5 | $x^{14} + 7 x^{3} + 7 x^{2} + 7$ | $14$ | $1$ | $15$ | $F_7$ | $[7/6]_{6}$ |