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Properties

Label 26.0.161...184.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.620\times 10^{49}$
Root discriminant \(78.10\)
Ramified primes $2,53$
Class number $14209$ (GRH)
Class group [14209] (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1)
 
gp: K = bnfinit(y^26 + 49*y^24 + 994*y^22 + 10915*y^20 + 71324*y^18 + 287620*y^16 + 721007*y^14 + 1111118*y^12 + 1019820*y^10 + 521449*y^8 + 129173*y^6 + 10874*y^4 + 236*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1)
 

\( x^{26} + 49 x^{24} + 994 x^{22} + 10915 x^{20} + 71324 x^{18} + 287620 x^{16} + 721007 x^{14} + 1111118 x^{12} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

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:   \(-16195738565069746760238624235485184480969630941184\) \(\medspace = -\,2^{26}\cdot 53^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(78.10\)
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:  $2\cdot 53^{12/13}\approx 78.10327769094482$
Ramified primes:   \(2\), \(53\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(212=2^{2}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{212}(1,·)$, $\chi_{212}(195,·)$, $\chi_{212}(69,·)$, $\chi_{212}(47,·)$, $\chi_{212}(203,·)$, $\chi_{212}(13,·)$, $\chi_{212}(77,·)$, $\chi_{212}(15,·)$, $\chi_{212}(81,·)$, $\chi_{212}(205,·)$, $\chi_{212}(89,·)$, $\chi_{212}(153,·)$, $\chi_{212}(201,·)$, $\chi_{212}(155,·)$, $\chi_{212}(95,·)$, $\chi_{212}(97,·)$, $\chi_{212}(99,·)$, $\chi_{212}(119,·)$, $\chi_{212}(169,·)$, $\chi_{212}(107,·)$, $\chi_{212}(175,·)$, $\chi_{212}(49,·)$, $\chi_{212}(183,·)$, $\chi_{212}(121,·)$, $\chi_{212}(187,·)$, $\chi_{212}(63,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{4096}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23}a^{16}-\frac{8}{23}a^{14}-\frac{7}{23}a^{10}-\frac{1}{23}a^{8}-\frac{4}{23}a^{6}+\frac{3}{23}a^{4}-\frac{8}{23}a^{2}-\frac{7}{23}$, $\frac{1}{23}a^{17}-\frac{8}{23}a^{15}-\frac{7}{23}a^{11}-\frac{1}{23}a^{9}-\frac{4}{23}a^{7}+\frac{3}{23}a^{5}-\frac{8}{23}a^{3}-\frac{7}{23}a$, $\frac{1}{23}a^{18}+\frac{5}{23}a^{14}-\frac{7}{23}a^{12}-\frac{11}{23}a^{10}+\frac{11}{23}a^{8}-\frac{6}{23}a^{6}-\frac{7}{23}a^{4}-\frac{2}{23}a^{2}-\frac{10}{23}$, $\frac{1}{23}a^{19}+\frac{5}{23}a^{15}-\frac{7}{23}a^{13}-\frac{11}{23}a^{11}+\frac{11}{23}a^{9}-\frac{6}{23}a^{7}-\frac{7}{23}a^{5}-\frac{2}{23}a^{3}-\frac{10}{23}a$, $\frac{1}{23}a^{20}+\frac{10}{23}a^{14}-\frac{11}{23}a^{12}-\frac{1}{23}a^{8}-\frac{10}{23}a^{6}+\frac{6}{23}a^{4}+\frac{7}{23}a^{2}-\frac{11}{23}$, $\frac{1}{23}a^{21}+\frac{10}{23}a^{15}-\frac{11}{23}a^{13}-\frac{1}{23}a^{9}-\frac{10}{23}a^{7}+\frac{6}{23}a^{5}+\frac{7}{23}a^{3}-\frac{11}{23}a$, $\frac{1}{4698049}a^{22}+\frac{34636}{4698049}a^{20}-\frac{91025}{4698049}a^{18}+\frac{26821}{4698049}a^{16}+\frac{379524}{4698049}a^{14}-\frac{255456}{4698049}a^{12}+\frac{2052906}{4698049}a^{10}+\frac{76757}{4698049}a^{8}+\frac{91766}{204263}a^{6}-\frac{1859271}{4698049}a^{4}+\frac{1415985}{4698049}a^{2}+\frac{2119684}{4698049}$, $\frac{1}{4698049}a^{23}+\frac{34636}{4698049}a^{21}-\frac{91025}{4698049}a^{19}+\frac{26821}{4698049}a^{17}+\frac{379524}{4698049}a^{15}-\frac{255456}{4698049}a^{13}+\frac{2052906}{4698049}a^{11}+\frac{76757}{4698049}a^{9}+\frac{91766}{204263}a^{7}-\frac{1859271}{4698049}a^{5}+\frac{1415985}{4698049}a^{3}+\frac{2119684}{4698049}a$, $\frac{1}{1070793422227}a^{24}+\frac{13278}{1070793422227}a^{22}-\frac{14186684266}{1070793422227}a^{20}-\frac{8086400107}{1070793422227}a^{18}-\frac{2421860596}{1070793422227}a^{16}+\frac{358322083704}{1070793422227}a^{14}-\frac{394474164910}{1070793422227}a^{12}-\frac{207849877869}{1070793422227}a^{10}-\frac{85443923132}{1070793422227}a^{8}+\frac{42726301558}{1070793422227}a^{6}+\frac{316600844179}{1070793422227}a^{4}+\frac{428682709255}{1070793422227}a^{2}-\frac{460404076292}{1070793422227}$, $\frac{1}{1070793422227}a^{25}+\frac{13278}{1070793422227}a^{23}-\frac{14186684266}{1070793422227}a^{21}-\frac{8086400107}{1070793422227}a^{19}-\frac{2421860596}{1070793422227}a^{17}+\frac{358322083704}{1070793422227}a^{15}-\frac{394474164910}{1070793422227}a^{13}-\frac{207849877869}{1070793422227}a^{11}-\frac{85443923132}{1070793422227}a^{9}+\frac{42726301558}{1070793422227}a^{7}+\frac{316600844179}{1070793422227}a^{5}+\frac{428682709255}{1070793422227}a^{3}-\frac{460404076292}{1070793422227}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{14209}$, which has order $14209$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{810188}{4698049} a^{25} - \frac{39731738}{4698049} a^{23} - \frac{806884047}{4698049} a^{21} - \frac{8873772064}{4698049} a^{19} - \frac{58106052589}{4698049} a^{17} - \frac{234977446694}{4698049} a^{15} - \frac{591220114790}{4698049} a^{13} - \frac{915134997367}{4698049} a^{11} - \frac{36668303943}{204263} a^{9} - \frac{431189338643}{4698049} a^{7} - \frac{104790832849}{4698049} a^{5} - \frac{7803798990}{4698049} a^{3} - \frac{4489009}{204263} a \)  (order $4$) Copy content Toggle raw display
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:   $\frac{11916336544}{1070793422227}a^{24}+\frac{595203583296}{1070793422227}a^{22}+\frac{115759325386}{10007415161}a^{20}+\frac{140708419307086}{1070793422227}a^{18}+\frac{961688141105570}{1070793422227}a^{16}+\frac{41\!\cdots\!94}{1070793422227}a^{14}+\frac{11\!\cdots\!12}{1070793422227}a^{12}+\frac{18\!\cdots\!46}{1070793422227}a^{10}+\frac{19\!\cdots\!45}{1070793422227}a^{8}+\frac{10\!\cdots\!92}{1070793422227}a^{6}+\frac{30\!\cdots\!14}{1070793422227}a^{4}+\frac{278890006931708}{1070793422227}a^{2}+\frac{4273730664724}{1070793422227}$, $\frac{7740934726}{1070793422227}a^{24}+\frac{384651799232}{1070793422227}a^{22}+\frac{7952410552536}{1070793422227}a^{20}+\frac{89612241637548}{1070793422227}a^{18}+\frac{606629964999181}{1070793422227}a^{16}+\frac{25\!\cdots\!54}{1070793422227}a^{14}+\frac{68\!\cdots\!50}{1070793422227}a^{12}+\frac{11\!\cdots\!26}{1070793422227}a^{10}+\frac{11\!\cdots\!68}{1070793422227}a^{8}+\frac{67\!\cdots\!86}{1070793422227}a^{6}+\frac{18\!\cdots\!46}{1070793422227}a^{4}+\frac{138751840293664}{1070793422227}a^{2}+\frac{1056472325532}{1070793422227}$, $\frac{28795710837}{1070793422227}a^{24}+\frac{1397958703327}{1070793422227}a^{22}+\frac{27993047778693}{1070793422227}a^{20}+\frac{301770501637293}{1070793422227}a^{18}+\frac{19\!\cdots\!03}{1070793422227}a^{16}+\frac{323576111130601}{46556235749}a^{14}+\frac{17\!\cdots\!90}{1070793422227}a^{12}+\frac{24\!\cdots\!56}{1070793422227}a^{10}+\frac{19\!\cdots\!19}{1070793422227}a^{8}+\frac{83\!\cdots\!35}{1070793422227}a^{6}+\frac{16\!\cdots\!29}{1070793422227}a^{4}+\frac{171852270672327}{1070793422227}a^{2}+\frac{3365120643142}{1070793422227}$, $\frac{28107656687}{1070793422227}a^{24}+\frac{1366209505447}{1070793422227}a^{22}+\frac{27407128599683}{1070793422227}a^{20}+\frac{12882223769871}{46556235749}a^{18}+\frac{82335315077210}{46556235749}a^{16}+\frac{73\!\cdots\!53}{1070793422227}a^{14}+\frac{17\!\cdots\!86}{1070793422227}a^{12}+\frac{11\!\cdots\!08}{46556235749}a^{10}+\frac{21\!\cdots\!13}{1070793422227}a^{8}+\frac{97\!\cdots\!66}{1070793422227}a^{6}+\frac{21\!\cdots\!33}{1070793422227}a^{4}+\frac{172241789884268}{1070793422227}a^{2}+\frac{2182566987738}{1070793422227}$, $\frac{400051796}{12901125569}a^{24}+\frac{70344559612}{46556235749}a^{22}+\frac{32564503183416}{1070793422227}a^{20}+\frac{353618932654976}{1070793422227}a^{18}+\frac{22\!\cdots\!59}{1070793422227}a^{16}+\frac{89\!\cdots\!64}{1070793422227}a^{14}+\frac{21\!\cdots\!09}{1070793422227}a^{12}+\frac{31\!\cdots\!00}{1070793422227}a^{10}+\frac{26\!\cdots\!48}{1070793422227}a^{8}+\frac{11\!\cdots\!77}{1070793422227}a^{6}+\frac{23\!\cdots\!97}{1070793422227}a^{4}+\frac{133203011664781}{1070793422227}a^{2}-\frac{1908306340310}{1070793422227}$, $\frac{857356692}{1070793422227}a^{24}+\frac{56237546751}{1070793422227}a^{22}+\frac{1541942002765}{1070793422227}a^{20}+\frac{23138868298505}{1070793422227}a^{18}+\frac{209166059758757}{1070793422227}a^{16}+\frac{11\!\cdots\!33}{1070793422227}a^{14}+\frac{42\!\cdots\!61}{1070793422227}a^{12}+\frac{92\!\cdots\!92}{1070793422227}a^{10}+\frac{11\!\cdots\!82}{1070793422227}a^{8}+\frac{84\!\cdots\!09}{1070793422227}a^{6}+\frac{26\!\cdots\!14}{1070793422227}a^{4}+\frac{214650490866441}{1070793422227}a^{2}+\frac{949552587358}{1070793422227}$, $\frac{20694233189}{1070793422227}a^{24}+\frac{1011293507922}{1070793422227}a^{22}+\frac{20439512461071}{1070793422227}a^{20}+\frac{9709102784094}{46556235749}a^{18}+\frac{62999773244876}{46556235749}a^{16}+\frac{57\!\cdots\!51}{1070793422227}a^{14}+\frac{14\!\cdots\!77}{1070793422227}a^{12}+\frac{941865028356279}{46556235749}a^{10}+\frac{19\!\cdots\!00}{1070793422227}a^{8}+\frac{96\!\cdots\!91}{1070793422227}a^{6}+\frac{23\!\cdots\!39}{1070793422227}a^{4}+\frac{192289249890871}{1070793422227}a^{2}+\frac{1296434045035}{1070793422227}$, $\frac{400051796}{12901125569}a^{24}+\frac{70344559612}{46556235749}a^{22}+\frac{32564503183416}{1070793422227}a^{20}+\frac{353618932654976}{1070793422227}a^{18}+\frac{22\!\cdots\!59}{1070793422227}a^{16}+\frac{89\!\cdots\!64}{1070793422227}a^{14}+\frac{21\!\cdots\!09}{1070793422227}a^{12}+\frac{31\!\cdots\!00}{1070793422227}a^{10}+\frac{26\!\cdots\!48}{1070793422227}a^{8}+\frac{11\!\cdots\!77}{1070793422227}a^{6}+\frac{23\!\cdots\!97}{1070793422227}a^{4}+\frac{134273805087008}{1070793422227}a^{2}+\frac{1304073926371}{1070793422227}$, $\frac{1904781418}{1070793422227}a^{24}+\frac{89947993380}{1070793422227}a^{22}+\frac{1728367796832}{1070793422227}a^{20}+\frac{17470409290813}{1070793422227}a^{18}+\frac{99808203574835}{1070793422227}a^{16}+\frac{316155053456076}{1070793422227}a^{14}+\frac{461888963140400}{1070793422227}a^{12}-\frac{82484868991007}{1070793422227}a^{10}-\frac{12\!\cdots\!66}{1070793422227}a^{8}-\frac{16\!\cdots\!48}{1070793422227}a^{6}-\frac{887249027535625}{1070793422227}a^{4}-\frac{174276606526896}{1070793422227}a^{2}-\frac{11043244491}{10007415161}$, $\frac{38057846408}{1070793422227}a^{24}+\frac{1874755454206}{1070793422227}a^{22}+\frac{38307850694506}{1070793422227}a^{20}+\frac{424887561367302}{1070793422227}a^{18}+\frac{28\!\cdots\!98}{1070793422227}a^{16}+\frac{11\!\cdots\!34}{1070793422227}a^{14}+\frac{29\!\cdots\!18}{1070793422227}a^{12}+\frac{47\!\cdots\!32}{1070793422227}a^{10}+\frac{45\!\cdots\!84}{1070793422227}a^{8}+\frac{24\!\cdots\!86}{1070793422227}a^{6}+\frac{62\!\cdots\!05}{1070793422227}a^{4}+\frac{468279065414314}{1070793422227}a^{2}+\frac{6315096059896}{1070793422227}$, $\frac{38915203100}{1070793422227}a^{24}+\frac{1930993000957}{1070793422227}a^{22}+\frac{39849792697271}{1070793422227}a^{20}+\frac{448026429665807}{1070793422227}a^{18}+\frac{30\!\cdots\!55}{1070793422227}a^{16}+\frac{12\!\cdots\!67}{1070793422227}a^{14}+\frac{34\!\cdots\!79}{1070793422227}a^{12}+\frac{56\!\cdots\!24}{1070793422227}a^{10}+\frac{57\!\cdots\!66}{1070793422227}a^{8}+\frac{32\!\cdots\!95}{1070793422227}a^{6}+\frac{88\!\cdots\!19}{1070793422227}a^{4}+\frac{682929556280755}{1070793422227}a^{2}+\frac{6193855225027}{1070793422227}$, $\frac{1209688347}{1070793422227}a^{24}+\frac{53410129381}{1070793422227}a^{22}+\frac{40262390149}{46556235749}a^{20}+\frac{7896776632688}{1070793422227}a^{18}+\frac{32453999950911}{1070793422227}a^{16}+\frac{35407532730054}{1070793422227}a^{14}-\frac{176912180891811}{1070793422227}a^{12}-\frac{599102072402421}{1070793422227}a^{10}-\frac{477581480919951}{1070793422227}a^{8}+\frac{346365847967809}{1070793422227}a^{6}+\frac{626451454778720}{1070793422227}a^{4}+\frac{217477066558669}{1070793422227}a^{2}+\frac{1518741153366}{1070793422227}$ Copy content Toggle raw display (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:  \( 5382739421.971964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 14209}{4\cdot\sqrt{16195738565069746760238624235485184480969630941184}}\cr\approx \mathstrut & 0.113017263873654 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{26}$ (as 26T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 13.13.491258904256726154641.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-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 $26$ ${\href{/padicField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/13.13.0.1}{13} }^{2}$ ${\href{/padicField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/23.2.0.1}{2} }^{13}$ ${\href{/padicField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/37.13.0.1}{13} }^{2}$ ${\href{/padicField/41.13.0.1}{13} }^{2}$ $26$ $26$ R $26$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $26$$2$$13$$26$
\(53\) Copy content Toggle raw display 53.13.12.1$x^{13} + 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$
53.13.12.1$x^{13} + 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$