Normalized defining polynomial
\( x^{14} + 4x^{12} - 348x^{10} + 260x^{8} + 209948x^{6} + 3803928x^{4} + 28024480x^{2} + 79121308 \)
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: | \(-94325144019389246216994698931499008\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 23^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(314.91\) | 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: | $2^{6/7}3^{6/7}7^{5/6}23^{6/7}\approx 345.4859899947601$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(23\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{52}a^{8}-\frac{3}{13}a^{6}-\frac{1}{2}a^{5}+\frac{5}{26}a^{4}-\frac{1}{2}a-\frac{3}{13}$, $\frac{1}{52}a^{9}+\frac{1}{52}a^{7}+\frac{5}{26}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{13}a-\frac{1}{2}$, $\frac{1}{52}a^{10}+\frac{11}{26}a^{6}+\frac{4}{13}a^{4}-\frac{1}{2}a^{3}-\frac{3}{13}a^{2}+\frac{3}{13}$, $\frac{1}{52}a^{11}-\frac{1}{13}a^{7}+\frac{4}{13}a^{5}-\frac{1}{2}a^{4}-\frac{3}{13}a^{3}+\frac{3}{13}a$, $\frac{1}{247586444956084}a^{12}-\frac{2046286492137}{247586444956084}a^{10}+\frac{2318785929131}{247586444956084}a^{8}-\frac{11937820009073}{123793222478042}a^{6}-\frac{54372576742405}{123793222478042}a^{4}-\frac{1}{2}a^{3}-\frac{17597616682774}{61896611239021}a^{2}-\frac{1}{2}a+\frac{9763103508211}{61896611239021}$, $\frac{1}{41\!\cdots\!04}a^{13}-\frac{651616309079659}{10\!\cdots\!01}a^{11}+\frac{12\!\cdots\!47}{20\!\cdots\!02}a^{9}+\frac{32\!\cdots\!33}{41\!\cdots\!04}a^{7}+\frac{31\!\cdots\!05}{10\!\cdots\!01}a^{5}-\frac{1}{2}a^{4}+\frac{57\!\cdots\!07}{20\!\cdots\!02}a^{3}-\frac{15\!\cdots\!29}{10\!\cdots\!01}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$ (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: | $\frac{36\!\cdots\!29}{123793222478042}a^{12}-\frac{78\!\cdots\!36}{61896611239021}a^{10}-\frac{59\!\cdots\!38}{61896611239021}a^{8}+\frac{56\!\cdots\!09}{61896611239021}a^{6}+\frac{34\!\cdots\!35}{61896611239021}a^{4}+\frac{40\!\cdots\!34}{61896611239021}a^{2}+\frac{14\!\cdots\!85}{61896611239021}$, $\frac{14\!\cdots\!59}{41\!\cdots\!04}a^{13}-\frac{62\!\cdots\!75}{123793222478042}a^{12}-\frac{94\!\cdots\!61}{41\!\cdots\!04}a^{11}+\frac{81\!\cdots\!57}{247586444956084}a^{10}-\frac{40\!\cdots\!29}{41\!\cdots\!04}a^{9}+\frac{17\!\cdots\!31}{123793222478042}a^{8}+\frac{23\!\cdots\!73}{20\!\cdots\!02}a^{7}-\frac{20\!\cdots\!23}{123793222478042}a^{6}+\frac{12\!\cdots\!49}{20\!\cdots\!02}a^{5}-\frac{11\!\cdots\!73}{123793222478042}a^{4}+\frac{13\!\cdots\!43}{20\!\cdots\!02}a^{3}-\frac{12\!\cdots\!01}{123793222478042}a^{2}+\frac{26\!\cdots\!59}{10\!\cdots\!01}a-\frac{23\!\cdots\!00}{61896611239021}$, $\frac{14\!\cdots\!59}{41\!\cdots\!04}a^{13}+\frac{62\!\cdots\!75}{123793222478042}a^{12}-\frac{94\!\cdots\!61}{41\!\cdots\!04}a^{11}-\frac{81\!\cdots\!57}{247586444956084}a^{10}-\frac{40\!\cdots\!29}{41\!\cdots\!04}a^{9}-\frac{17\!\cdots\!31}{123793222478042}a^{8}+\frac{23\!\cdots\!73}{20\!\cdots\!02}a^{7}+\frac{20\!\cdots\!23}{123793222478042}a^{6}+\frac{12\!\cdots\!49}{20\!\cdots\!02}a^{5}+\frac{11\!\cdots\!73}{123793222478042}a^{4}+\frac{13\!\cdots\!43}{20\!\cdots\!02}a^{3}+\frac{12\!\cdots\!01}{123793222478042}a^{2}+\frac{26\!\cdots\!59}{10\!\cdots\!01}a+\frac{23\!\cdots\!00}{61896611239021}$, $\frac{22\!\cdots\!99}{41\!\cdots\!04}a^{13}+\frac{31\!\cdots\!39}{247586444956084}a^{12}-\frac{57\!\cdots\!67}{41\!\cdots\!04}a^{11}-\frac{12\!\cdots\!98}{61896611239021}a^{10}-\frac{23\!\cdots\!48}{10\!\cdots\!01}a^{9}-\frac{34\!\cdots\!17}{123793222478042}a^{8}+\frac{46\!\cdots\!97}{20\!\cdots\!02}a^{7}+\frac{11\!\cdots\!43}{123793222478042}a^{6}+\frac{22\!\cdots\!21}{20\!\cdots\!02}a^{5}+\frac{17\!\cdots\!67}{123793222478042}a^{4}+\frac{10\!\cdots\!40}{10\!\cdots\!01}a^{3}+\frac{28\!\cdots\!74}{61896611239021}a^{2}+\frac{33\!\cdots\!43}{10\!\cdots\!01}a-\frac{46\!\cdots\!99}{61896611239021}$, $\frac{15\!\cdots\!71}{41\!\cdots\!04}a^{13}-\frac{26\!\cdots\!63}{247586444956084}a^{12}-\frac{56\!\cdots\!61}{10\!\cdots\!01}a^{11}+\frac{18\!\cdots\!79}{247586444956084}a^{10}-\frac{57\!\cdots\!35}{41\!\cdots\!04}a^{9}+\frac{19\!\cdots\!90}{61896611239021}a^{8}+\frac{19\!\cdots\!87}{20\!\cdots\!02}a^{7}-\frac{49\!\cdots\!03}{123793222478042}a^{6}+\frac{80\!\cdots\!07}{10\!\cdots\!01}a^{5}-\frac{11\!\cdots\!59}{61896611239021}a^{4}+\frac{20\!\cdots\!39}{20\!\cdots\!02}a^{3}-\frac{23\!\cdots\!19}{123793222478042}a^{2}+\frac{43\!\cdots\!94}{10\!\cdots\!01}a-\frac{34\!\cdots\!87}{61896611239021}$, $\frac{15\!\cdots\!71}{41\!\cdots\!04}a^{13}+\frac{26\!\cdots\!63}{247586444956084}a^{12}-\frac{56\!\cdots\!61}{10\!\cdots\!01}a^{11}-\frac{18\!\cdots\!79}{247586444956084}a^{10}-\frac{57\!\cdots\!35}{41\!\cdots\!04}a^{9}-\frac{19\!\cdots\!90}{61896611239021}a^{8}+\frac{19\!\cdots\!87}{20\!\cdots\!02}a^{7}+\frac{49\!\cdots\!03}{123793222478042}a^{6}+\frac{80\!\cdots\!07}{10\!\cdots\!01}a^{5}+\frac{11\!\cdots\!59}{61896611239021}a^{4}+\frac{20\!\cdots\!39}{20\!\cdots\!02}a^{3}+\frac{23\!\cdots\!19}{123793222478042}a^{2}+\frac{43\!\cdots\!94}{10\!\cdots\!01}a+\frac{34\!\cdots\!87}{61896611239021}$ (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: | \( 526166190630.4457 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 526166190630.4457 \cdot 7}{2\cdot\sqrt{94325144019389246216994698931499008}}\cr\approx \mathstrut & 2.31812218325358 \end{aligned}\] (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
\(\Q(\sqrt{-7}) \), 7.1.116081956281751488.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 42 |
Degree 7 sibling: | 7.1.116081956281751488.2 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.116081956281751488.2 |
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 | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.1.0.1}{1} }^{14}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(23\) | 23.7.6.1 | $x^{7} + 23$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
23.7.6.1 | $x^{7} + 23$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |