Normalized defining polynomial
\( x^{26} + 6 x^{24} + 189 x^{22} + 2700 x^{20} + 22032 x^{18} + 262683 x^{16} + 3276126 x^{14} + \cdots + 419306949 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3081633091018542230201228818947762663997833216\) \(\medspace = -\,2^{26}\cdot 3^{13}\cdot 263^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(56.18\) | 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: | $2\cdot 3^{1/2}263^{1/2}\approx 56.178287620752556$ | ||
Ramified primes: | \(2\), \(3\), \(263\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-789}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{10935}a^{14}-\frac{1}{3645}a^{12}+\frac{1}{1215}a^{10}+\frac{2}{405}a^{8}+\frac{2}{45}a^{4}-\frac{1}{5}$, $\frac{1}{10935}a^{15}-\frac{1}{3645}a^{13}+\frac{1}{1215}a^{11}+\frac{2}{405}a^{9}+\frac{2}{45}a^{5}-\frac{1}{5}a$, $\frac{1}{32805}a^{16}-\frac{2}{1215}a^{10}+\frac{2}{405}a^{8}+\frac{2}{135}a^{6}+\frac{2}{45}a^{4}-\frac{1}{15}a^{2}-\frac{1}{5}$, $\frac{1}{32805}a^{17}-\frac{2}{1215}a^{11}+\frac{2}{405}a^{9}+\frac{2}{135}a^{7}+\frac{2}{45}a^{5}-\frac{1}{15}a^{3}-\frac{1}{5}a$, $\frac{1}{98415}a^{18}-\frac{2}{3645}a^{12}+\frac{2}{1215}a^{10}+\frac{2}{405}a^{8}+\frac{2}{135}a^{6}-\frac{1}{45}a^{4}-\frac{1}{15}a^{2}$, $\frac{1}{98415}a^{19}-\frac{2}{3645}a^{13}+\frac{2}{1215}a^{11}+\frac{2}{405}a^{9}+\frac{2}{135}a^{7}-\frac{1}{45}a^{5}-\frac{1}{15}a^{3}$, $\frac{1}{295245}a^{20}-\frac{1}{1215}a^{10}+\frac{1}{405}a^{8}-\frac{1}{135}a^{6}-\frac{2}{45}a^{4}-\frac{2}{5}$, $\frac{1}{295245}a^{21}-\frac{1}{1215}a^{11}+\frac{1}{405}a^{9}-\frac{1}{135}a^{7}-\frac{2}{45}a^{5}-\frac{2}{5}a$, $\frac{1}{4428675}a^{22}+\frac{2}{54675}a^{14}+\frac{2}{18225}a^{12}-\frac{2}{6075}a^{10}-\frac{2}{2025}a^{8}-\frac{7}{675}a^{6}+\frac{4}{225}a^{4}-\frac{4}{25}a^{2}+\frac{3}{25}$, $\frac{1}{4428675}a^{23}+\frac{2}{54675}a^{15}+\frac{2}{18225}a^{13}-\frac{2}{6075}a^{11}-\frac{2}{2025}a^{9}-\frac{7}{675}a^{7}+\frac{4}{225}a^{5}-\frac{4}{25}a^{3}+\frac{3}{25}a$, $\frac{1}{90\!\cdots\!75}a^{24}-\frac{11\!\cdots\!59}{30\!\cdots\!25}a^{22}-\frac{33\!\cdots\!56}{20\!\cdots\!75}a^{20}-\frac{11\!\cdots\!69}{67\!\cdots\!25}a^{18}+\frac{21\!\cdots\!56}{16\!\cdots\!25}a^{16}-\frac{17\!\cdots\!06}{37\!\cdots\!25}a^{14}+\frac{37\!\cdots\!28}{27\!\cdots\!75}a^{12}-\frac{26\!\cdots\!64}{41\!\cdots\!25}a^{10}+\frac{58\!\cdots\!16}{13\!\cdots\!75}a^{8}-\frac{23\!\cdots\!96}{15\!\cdots\!75}a^{6}+\frac{74\!\cdots\!07}{15\!\cdots\!75}a^{4}-\frac{46\!\cdots\!34}{51\!\cdots\!25}a^{2}-\frac{97\!\cdots\!92}{17\!\cdots\!75}$, $\frac{1}{90\!\cdots\!75}a^{25}-\frac{11\!\cdots\!59}{30\!\cdots\!25}a^{23}-\frac{33\!\cdots\!56}{20\!\cdots\!75}a^{21}-\frac{11\!\cdots\!69}{67\!\cdots\!25}a^{19}+\frac{21\!\cdots\!56}{16\!\cdots\!25}a^{17}-\frac{17\!\cdots\!06}{37\!\cdots\!25}a^{15}+\frac{37\!\cdots\!28}{27\!\cdots\!75}a^{13}-\frac{26\!\cdots\!64}{41\!\cdots\!25}a^{11}+\frac{58\!\cdots\!16}{13\!\cdots\!75}a^{9}-\frac{23\!\cdots\!96}{15\!\cdots\!75}a^{7}+\frac{74\!\cdots\!07}{15\!\cdots\!75}a^{5}-\frac{46\!\cdots\!34}{51\!\cdots\!25}a^{3}-\frac{97\!\cdots\!92}{17\!\cdots\!75}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
not computed
Unit group
Rank: | $12$ | 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: | not computed | 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: | not computed | 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 $
Galois group
A solvable group of order 52 |
The 16 conjugacy class representatives for $D_{26}$ |
Character table for $D_{26}$ |
Intermediate fields
\(\Q(\sqrt{-789}) \), 13.1.330928743953809.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 26 sibling: | 26.2.11717236087522974259320261669002899863109632.1 |
Minimal sibling: | 26.2.11717236087522974259320261669002899863109632.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{12}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.13.0.1}{13} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.13.0.1}{13} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/47.2.0.1}{2} }^{13}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{13}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $26$ | $2$ | $13$ | $26$ | |||
\(3\) | Deg $26$ | $2$ | $13$ | $13$ | |||
\(263\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.3156.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 263 $ | \(\Q(\sqrt{-789}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.263.2t1.a.a | $1$ | $ 263 $ | \(\Q(\sqrt{-263}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.37872.26t3.a.f | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.263.13t2.a.c | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.263.13t2.a.b | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.37872.26t3.a.e | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.37872.26t3.a.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.263.13t2.a.a | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.263.13t2.a.d | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.263.13t2.a.f | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.37872.26t3.a.d | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.37872.26t3.a.c | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
* | 2.263.13t2.a.e | $2$ | $ 263 $ | 13.1.330928743953809.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
* | 2.37872.26t3.a.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 263 $ | 26.0.3081633091018542230201228818947762663997833216.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |