Normalized defining polynomial
\( x^{14} - 59x^{12} + 1269x^{10} - 11395x^{8} + 28655x^{6} + 64563x^{4} + 245323x^{2} + 514087 \)
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: | \(-1050832501626663000000000000\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{11}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(85.14\) | 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: | $2^{6/7}3^{6/7}5^{6/7}7^{5/6}\approx 93.40116629955774$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{35553962804588}a^{12}+\frac{1010920079589}{17776981402294}a^{10}+\frac{3196826415993}{35553962804588}a^{8}-\frac{3053237149206}{8888490701147}a^{6}-\frac{1}{2}a^{5}-\frac{3927759735011}{35553962804588}a^{4}+\frac{8034878819275}{17776981402294}a^{2}-\frac{1}{2}a-\frac{13716068369371}{35553962804588}$, $\frac{1}{96\!\cdots\!48}a^{13}+\frac{178780734102529}{48\!\cdots\!74}a^{11}-\frac{370119783032181}{96\!\cdots\!48}a^{9}+\frac{592475639827643}{24\!\cdots\!37}a^{7}-\frac{1}{2}a^{6}+\frac{22\!\cdots\!21}{96\!\cdots\!48}a^{5}-\frac{19\!\cdots\!53}{48\!\cdots\!74}a^{3}-\frac{1}{2}a^{2}-\frac{45\!\cdots\!41}{96\!\cdots\!48}a$
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)
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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: | $\frac{17910040930409}{96\!\cdots\!48}a^{13}-\frac{104310982125}{17776981402294}a^{12}-\frac{888427804281183}{96\!\cdots\!48}a^{11}+\frac{2644993859292}{8888490701147}a^{10}+\frac{34\!\cdots\!99}{24\!\cdots\!37}a^{9}-\frac{85082627892371}{17776981402294}a^{8}-\frac{14\!\cdots\!76}{24\!\cdots\!37}a^{7}+\frac{182955991690851}{8888490701147}a^{6}-\frac{14\!\cdots\!11}{96\!\cdots\!48}a^{5}+\frac{900388645409389}{17776981402294}a^{4}-\frac{46\!\cdots\!91}{96\!\cdots\!48}a^{3}+\frac{29\!\cdots\!39}{17776981402294}a^{2}-\frac{20\!\cdots\!87}{24\!\cdots\!37}a+\frac{25\!\cdots\!76}{8888490701147}$, $\frac{17910040930409}{96\!\cdots\!48}a^{13}+\frac{104310982125}{17776981402294}a^{12}-\frac{888427804281183}{96\!\cdots\!48}a^{11}-\frac{2644993859292}{8888490701147}a^{10}+\frac{34\!\cdots\!99}{24\!\cdots\!37}a^{9}+\frac{85082627892371}{17776981402294}a^{8}-\frac{14\!\cdots\!76}{24\!\cdots\!37}a^{7}-\frac{182955991690851}{8888490701147}a^{6}-\frac{14\!\cdots\!11}{96\!\cdots\!48}a^{5}-\frac{900388645409389}{17776981402294}a^{4}-\frac{46\!\cdots\!91}{96\!\cdots\!48}a^{3}-\frac{29\!\cdots\!39}{17776981402294}a^{2}-\frac{20\!\cdots\!87}{24\!\cdots\!37}a-\frac{25\!\cdots\!76}{8888490701147}$, $\frac{7456827236945}{48\!\cdots\!74}a^{13}+\frac{256081842397}{35553962804588}a^{12}-\frac{142562829668723}{24\!\cdots\!37}a^{11}-\frac{10140663002093}{35553962804588}a^{10}+\frac{32\!\cdots\!11}{48\!\cdots\!74}a^{9}+\frac{62235978856879}{17776981402294}a^{8}-\frac{28\!\cdots\!98}{24\!\cdots\!37}a^{7}-\frac{174908127720225}{17776981402294}a^{6}-\frac{24\!\cdots\!30}{24\!\cdots\!37}a^{5}-\frac{14\!\cdots\!85}{35553962804588}a^{4}-\frac{23\!\cdots\!11}{48\!\cdots\!74}a^{3}-\frac{45\!\cdots\!15}{35553962804588}a^{2}-\frac{33\!\cdots\!79}{48\!\cdots\!74}a-\frac{16\!\cdots\!17}{8888490701147}$, $\frac{11341325944359}{96\!\cdots\!48}a^{13}-\frac{5940996234}{8888490701147}a^{12}-\frac{715108702516267}{96\!\cdots\!48}a^{11}+\frac{904735731805}{17776981402294}a^{10}+\frac{42\!\cdots\!43}{24\!\cdots\!37}a^{9}-\frac{12490087424583}{8888490701147}a^{8}-\frac{43\!\cdots\!02}{24\!\cdots\!37}a^{7}+\frac{147838207934130}{8888490701147}a^{6}+\frac{69\!\cdots\!71}{96\!\cdots\!48}a^{5}-\frac{628381588525201}{8888490701147}a^{4}-\frac{35\!\cdots\!11}{96\!\cdots\!48}a^{3}+\frac{187592923601991}{8888490701147}a^{2}+\frac{94\!\cdots\!42}{24\!\cdots\!37}a-\frac{70\!\cdots\!59}{17776981402294}$, $\frac{11341325944359}{96\!\cdots\!48}a^{13}+\frac{5940996234}{8888490701147}a^{12}-\frac{715108702516267}{96\!\cdots\!48}a^{11}-\frac{904735731805}{17776981402294}a^{10}+\frac{42\!\cdots\!43}{24\!\cdots\!37}a^{9}+\frac{12490087424583}{8888490701147}a^{8}-\frac{43\!\cdots\!02}{24\!\cdots\!37}a^{7}-\frac{147838207934130}{8888490701147}a^{6}+\frac{69\!\cdots\!71}{96\!\cdots\!48}a^{5}+\frac{628381588525201}{8888490701147}a^{4}-\frac{35\!\cdots\!11}{96\!\cdots\!48}a^{3}-\frac{187592923601991}{8888490701147}a^{2}+\frac{94\!\cdots\!42}{24\!\cdots\!37}a+\frac{70\!\cdots\!59}{17776981402294}$, $\frac{165774083945}{48\!\cdots\!74}a^{13}-\frac{14211408013}{8888490701147}a^{12}-\frac{24272085009261}{96\!\cdots\!48}a^{11}+\frac{3304283937939}{35553962804588}a^{10}+\frac{157953244321972}{24\!\cdots\!37}a^{9}-\frac{17387732267731}{8888490701147}a^{8}-\frac{15\!\cdots\!79}{24\!\cdots\!37}a^{7}+\frac{301588775336301}{17776981402294}a^{6}+\frac{472687505418921}{24\!\cdots\!37}a^{5}-\frac{662157810274929}{17776981402294}a^{4}+\frac{17\!\cdots\!13}{96\!\cdots\!48}a^{3}-\frac{55\!\cdots\!95}{35553962804588}a^{2}+\frac{25\!\cdots\!75}{48\!\cdots\!74}a+\frac{368195611687163}{8888490701147}$ (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: | \( 237529515.95679548 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 237529515.95679548 \cdot 1}{2\cdot\sqrt{1050832501626663000000000000}}\cr\approx \mathstrut & 1.41638043214802 \end{aligned}\] (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
\(\Q(\sqrt{-7}) \), 7.1.12252303000000.8 |
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: | 7.1.12252303000000.8 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.12252303000000.8 |
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 | R | 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 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(5\) | 5.14.12.1 | $x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |