Normalized defining polynomial
\( x^{45} - 3 x^{44} + 19 x^{43} - 46 x^{42} + 57 x^{41} - 164 x^{40} - 1542 x^{39} + 2752 x^{38} + \cdots - 879691 \)
Invariants
| Degree: | $45$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-226\!\cdots\!891\)
\(\medspace = -\,11^{39}\cdot 31^{30}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(78.85\) | 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: | $11^{9/10}31^{2/3}\approx 85.40721213930478$ | ||
| Ramified primes: |
\(11\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $15$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{19}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{18}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{19}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{35}-\frac{1}{4}a^{34}-\frac{1}{4}a^{33}-\frac{1}{4}a^{30}-\frac{1}{4}a^{28}-\frac{1}{4}a^{27}-\frac{1}{4}a^{25}-\frac{1}{4}a^{24}+\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}-\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{36}-\frac{1}{4}a^{33}-\frac{1}{4}a^{31}-\frac{1}{4}a^{30}-\frac{1}{4}a^{29}-\frac{1}{4}a^{27}-\frac{1}{4}a^{26}-\frac{1}{4}a^{24}-\frac{1}{4}a^{20}-\frac{1}{2}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}$, $\frac{1}{4}a^{37}-\frac{1}{4}a^{34}-\frac{1}{4}a^{32}-\frac{1}{4}a^{31}-\frac{1}{4}a^{30}-\frac{1}{4}a^{28}-\frac{1}{4}a^{27}-\frac{1}{4}a^{25}-\frac{1}{4}a^{21}-\frac{1}{4}a^{19}-\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{38}-\frac{1}{4}a^{34}-\frac{1}{4}a^{32}-\frac{1}{4}a^{31}-\frac{1}{4}a^{30}-\frac{1}{4}a^{29}-\frac{1}{4}a^{27}-\frac{1}{4}a^{26}-\frac{1}{4}a^{25}-\frac{1}{4}a^{24}-\frac{1}{4}a^{22}-\frac{1}{4}a^{20}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{2}a^{13}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{39}-\frac{1}{4}a^{34}-\frac{1}{4}a^{32}-\frac{1}{4}a^{31}-\frac{1}{4}a^{26}-\frac{1}{4}a^{24}-\frac{1}{4}a^{23}-\frac{1}{4}a^{21}-\frac{1}{4}a^{19}+\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{40}-\frac{1}{8}a^{39}-\frac{1}{8}a^{36}-\frac{1}{4}a^{34}-\frac{1}{8}a^{33}-\frac{1}{4}a^{31}-\frac{1}{8}a^{29}-\frac{1}{8}a^{28}+\frac{1}{8}a^{27}-\frac{1}{4}a^{25}+\frac{1}{8}a^{23}-\frac{1}{8}a^{22}+\frac{1}{8}a^{21}-\frac{1}{8}a^{19}-\frac{3}{8}a^{16}+\frac{3}{8}a^{15}-\frac{1}{2}a^{14}+\frac{3}{8}a^{13}+\frac{3}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{3}{8}a^{8}+\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{8}$, $\frac{1}{8}a^{41}-\frac{1}{8}a^{39}-\frac{1}{8}a^{37}-\frac{1}{8}a^{36}-\frac{1}{8}a^{34}+\frac{1}{8}a^{33}-\frac{1}{4}a^{32}-\frac{1}{4}a^{31}+\frac{1}{8}a^{30}-\frac{1}{4}a^{29}-\frac{1}{4}a^{28}-\frac{1}{8}a^{27}-\frac{1}{4}a^{26}-\frac{1}{8}a^{24}+\frac{1}{8}a^{21}-\frac{1}{8}a^{20}-\frac{3}{8}a^{19}+\frac{1}{4}a^{18}+\frac{1}{8}a^{17}+\frac{1}{4}a^{16}+\frac{3}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{3}{8}a^{11}+\frac{1}{4}a^{10}-\frac{3}{8}a^{9}-\frac{3}{8}a^{8}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{8}a^{42}-\frac{1}{8}a^{39}-\frac{1}{8}a^{38}-\frac{1}{8}a^{37}-\frac{1}{8}a^{36}-\frac{1}{8}a^{35}-\frac{1}{8}a^{34}+\frac{1}{8}a^{33}-\frac{1}{4}a^{32}-\frac{1}{8}a^{31}-\frac{1}{4}a^{30}+\frac{1}{8}a^{29}-\frac{1}{4}a^{28}-\frac{1}{8}a^{27}+\frac{1}{8}a^{25}+\frac{1}{8}a^{23}+\frac{1}{8}a^{20}-\frac{3}{8}a^{19}-\frac{3}{8}a^{18}+\frac{1}{4}a^{17}-\frac{1}{2}a^{16}-\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a-\frac{1}{8}$, $\frac{1}{186122296}a^{43}-\frac{6654831}{186122296}a^{42}+\frac{5684313}{186122296}a^{41}+\frac{88913}{1707544}a^{40}-\frac{9426497}{186122296}a^{39}-\frac{5774539}{46530574}a^{38}+\frac{925407}{186122296}a^{37}-\frac{5404423}{186122296}a^{36}-\frac{4013629}{93061148}a^{35}+\frac{11807627}{186122296}a^{34}+\frac{5221049}{23265287}a^{33}+\frac{19734491}{186122296}a^{32}-\frac{42928191}{186122296}a^{31}-\frac{435165}{93061148}a^{30}-\frac{18181521}{186122296}a^{29}-\frac{5171173}{186122296}a^{28}-\frac{21488739}{93061148}a^{27}+\frac{22879279}{186122296}a^{26}+\frac{11049193}{186122296}a^{25}-\frac{6971839}{46530574}a^{24}-\frac{11537301}{186122296}a^{23}+\frac{294089}{23265287}a^{22}-\frac{12102657}{93061148}a^{21}-\frac{22119807}{186122296}a^{20}+\frac{39882295}{186122296}a^{19}+\frac{19236941}{186122296}a^{18}+\frac{48522703}{186122296}a^{17}+\frac{30129091}{93061148}a^{16}+\frac{85284079}{186122296}a^{15}-\frac{8530033}{23265287}a^{14}-\frac{13508281}{186122296}a^{13}+\frac{52321073}{186122296}a^{12}-\frac{88561933}{186122296}a^{11}+\frac{16732021}{46530574}a^{10}-\frac{14564221}{93061148}a^{9}+\frac{32454935}{93061148}a^{8}+\frac{79833067}{186122296}a^{7}-\frac{17572705}{186122296}a^{6}-\frac{86016259}{186122296}a^{5}+\frac{28929859}{186122296}a^{4}+\frac{44837001}{93061148}a^{3}+\frac{478233}{46530574}a^{2}-\frac{38138335}{186122296}a+\frac{6217359}{93061148}$, $\frac{1}{16\!\cdots\!72}a^{44}-\frac{18\!\cdots\!09}{16\!\cdots\!72}a^{43}+\frac{46\!\cdots\!01}{40\!\cdots\!18}a^{42}+\frac{74\!\cdots\!57}{20\!\cdots\!59}a^{41}-\frac{38\!\cdots\!35}{80\!\cdots\!36}a^{40}-\frac{11\!\cdots\!65}{16\!\cdots\!72}a^{39}+\frac{75\!\cdots\!41}{80\!\cdots\!36}a^{38}-\frac{17\!\cdots\!81}{16\!\cdots\!72}a^{37}-\frac{38\!\cdots\!01}{16\!\cdots\!72}a^{36}-\frac{18\!\cdots\!53}{20\!\cdots\!59}a^{35}-\frac{14\!\cdots\!95}{80\!\cdots\!36}a^{34}-\frac{27\!\cdots\!49}{40\!\cdots\!18}a^{33}+\frac{19\!\cdots\!33}{16\!\cdots\!72}a^{32}-\frac{24\!\cdots\!41}{16\!\cdots\!72}a^{31}+\frac{26\!\cdots\!65}{40\!\cdots\!18}a^{30}+\frac{28\!\cdots\!41}{16\!\cdots\!72}a^{29}+\frac{39\!\cdots\!91}{16\!\cdots\!72}a^{28}-\frac{13\!\cdots\!99}{20\!\cdots\!59}a^{27}-\frac{17\!\cdots\!65}{16\!\cdots\!72}a^{26}-\frac{22\!\cdots\!23}{16\!\cdots\!72}a^{25}-\frac{60\!\cdots\!87}{80\!\cdots\!36}a^{24}-\frac{45\!\cdots\!05}{20\!\cdots\!59}a^{23}+\frac{13\!\cdots\!41}{16\!\cdots\!72}a^{22}-\frac{41\!\cdots\!51}{16\!\cdots\!72}a^{21}+\frac{12\!\cdots\!11}{16\!\cdots\!72}a^{20}+\frac{12\!\cdots\!31}{40\!\cdots\!18}a^{19}+\frac{25\!\cdots\!27}{80\!\cdots\!36}a^{18}-\frac{75\!\cdots\!53}{16\!\cdots\!72}a^{17}-\frac{19\!\cdots\!41}{80\!\cdots\!36}a^{16}+\frac{14\!\cdots\!41}{80\!\cdots\!36}a^{15}-\frac{15\!\cdots\!95}{80\!\cdots\!36}a^{14}-\frac{40\!\cdots\!13}{16\!\cdots\!72}a^{13}-\frac{99\!\cdots\!09}{40\!\cdots\!18}a^{12}-\frac{12\!\cdots\!23}{16\!\cdots\!72}a^{11}+\frac{61\!\cdots\!61}{16\!\cdots\!72}a^{10}+\frac{52\!\cdots\!63}{30\!\cdots\!72}a^{9}+\frac{49\!\cdots\!07}{16\!\cdots\!72}a^{8}+\frac{74\!\cdots\!43}{16\!\cdots\!72}a^{7}+\frac{11\!\cdots\!03}{40\!\cdots\!18}a^{6}+\frac{45\!\cdots\!37}{20\!\cdots\!59}a^{5}+\frac{47\!\cdots\!89}{16\!\cdots\!72}a^{4}-\frac{55\!\cdots\!49}{16\!\cdots\!72}a^{3}-\frac{50\!\cdots\!25}{16\!\cdots\!72}a^{2}-\frac{11\!\cdots\!25}{80\!\cdots\!36}a+\frac{72\!\cdots\!63}{16\!\cdots\!72}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $29$ | 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
$S_3\times C_{15}$ (as 45T3):
| A solvable group of order 90 |
| The 45 conjugacy class representatives for $S_3\times C_{15}$ |
| Character table for $S_3\times C_{15}$ |
Intermediate fields
| 3.3.961.1, 3.1.10571.1, \(\Q(\zeta_{11})^+\), 9.3.1181267399411.1, 15.15.2572344674223769220522333521.1, 15.5.28295791416461461425745668731.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 30.0.10745322081507339108891258824483901499337171.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{3}{,}\,{\href{/padicField/2.5.0.1}{5} }^{3}$ | $15^{3}$ | $15^{3}$ | $30{,}\,15$ | R | $30{,}\,15$ | $30{,}\,15$ | $30{,}\,15$ | ${\href{/padicField/23.3.0.1}{3} }^{15}$ | ${\href{/padicField/29.10.0.1}{10} }^{3}{,}\,{\href{/padicField/29.5.0.1}{5} }^{3}$ | R | $15^{3}$ | $30{,}\,15$ | ${\href{/padicField/43.6.0.1}{6} }^{5}{,}\,{\href{/padicField/43.3.0.1}{3} }^{5}$ | $15^{3}$ | $15^{3}$ | $15^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.15.12.1 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 393 x^{10} + 890 x^{9} + 750 x^{8} - 3970 x^{7} + 9610 x^{6} + 13085 x^{5} + 151045 x^{4} + 26525 x^{3} + 116170 x^{2} - 53795 x + 67662$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| Deg $30$ | $10$ | $3$ | $27$ | ||||
|
\(31\)
| Deg $45$ | $3$ | $15$ | $30$ |