↑

Learn more

Note: Search results may be incomplete. Given $p$-adic completions contain an unramified field and completions are only searched for ramified primes.

Refine search

Degree Signature Ramified prime count Ramified Unramified primes
Galois group Field is Intermediate field Class group structure Class number
Discriminant Root discriminant Galois root discriminant Narrow class group Narrow class number
CM field Relative class number Regulator $p$-adic completions Monogenic
Index Ines. \(p\) Minimal sibling
Sort order Select

Results (1-50 of at least 1000)

Next   To download results, determine the number of results.
Label Polynomial Discriminant Galois group Class group Regulator
4.0.320.1 $x^{4} - 2 x^{3} + 2$ $2^{6}\cdot 5$ $D_{4}$ (as 4T3) trivial $1.06127506191$
4.0.605.1 $x^{4} - x^{3} + x^{2} + x + 1$ $5\cdot 11^{2}$ $D_{4}$ (as 4T3) trivial $0.767197218251$
4.0.985.1 $x^{4} - x^{3} + 2 x^{2} - 3 x + 2$ $5\cdot 197$ $S_4$ (as 4T5) trivial $1.34361173004$
4.0.1040.2 $x^{4} + x^{2} - 4 x + 4$ $2^{4}\cdot 5\cdot 13$ $D_{4}$ (as 4T3) trivial $3.46864904298$
4.2.1255.1 $x^{4} - x^{2} - 3 x - 1$ $-\,5\cdot 251$ $S_4$ (as 4T5) trivial $1.58232994952$
4.0.1940.1 $x^{4} - x^{3} + 2$ $2^{2}\cdot 5\cdot 97$ $S_4$ (as 4T5) trivial $2.67428117567$
4.0.2320.1 $x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 8$ $2^{4}\cdot 5\cdot 29$ $D_{4}$ (as 4T3) trivial $5.95755332391$
4.0.2320.2 $x^{4} + x^{2} - 6 x + 9$ $2^{4}\cdot 5\cdot 29$ $D_{4}$ (as 4T3) trivial $2.5209503756$
4.2.2480.1 $x^{4} - 2 x - 2$ $-\,2^{4}\cdot 5\cdot 31$ $S_4$ (as 4T5) trivial $2.90991289264$
4.0.2560.2 $x^{4} - 2 x^{2} + 10$ $2^{9}\cdot 5$ $D_{4}$ (as 4T3) trivial $3.93727585159$
4.0.2920.1 $x^{4} - x^{2} - 2 x + 4$ $2^{3}\cdot 5\cdot 73$ $S_4$ (as 4T5) trivial $3.02461741979$
4.0.2960.2 $x^{4} - 5 x^{2} - 4 x + 13$ $2^{4}\cdot 5\cdot 37$ $D_{4}$ (as 4T3) trivial $2.72360180172$
4.0.3760.1 $x^{4} - 2 x^{3} - x^{2} + 5$ $2^{4}\cdot 5\cdot 47$ $S_4$ (as 4T5) trivial $2.16622673051$
4.4.4205.1 $x^{4} - x^{3} - 5 x^{2} - x + 1$ $5\cdot 29^{2}$ $D_{4}$ (as 4T3) trivial $2.8242845114$
4.0.4240.1 $x^{4} - 5 x^{2} - 6 x + 18$ $2^{4}\cdot 5\cdot 53$ $D_{4}$ (as 4T3) trivial $8.6189189872$
4.0.4345.1 $x^{4} - x^{3} + x^{2} - 2 x + 3$ $5\cdot 11\cdot 79$ $S_4$ (as 4T5) trivial $3.62072447578$
4.0.4360.1 $x^{4} - x^{3} + x^{2} + 2 x + 2$ $2^{3}\cdot 5\cdot 109$ $S_4$ (as 4T5) trivial $3.76508899038$
4.2.4595.1 $x^{4} - 2 x^{3} - 3 x + 1$ $-\,5\cdot 919$ $S_4$ (as 4T5) trivial $2.80979477624$
4.2.4615.1 $x^{4} - x^{3} + x^{2} - 4 x + 1$ $-\,5\cdot 13\cdot 71$ $S_4$ (as 4T5) trivial $3.61991207497$
4.2.4780.1 $x^{4} - x^{3} - 3 x^{2} - 3 x + 2$ $-\,2^{2}\cdot 5\cdot 239$ $S_4$ (as 4T5) trivial $5.67990769941$
4.0.4805.1 $x^{4} - 2 x^{3} + 2 x^{2} - x + 8$ $5\cdot 31^{2}$ $D_{4}$ (as 4T3) $[3]$ $1.80358267247$
4.2.5260.1 $x^{4} - x^{3} + x^{2} + 3 x - 2$ $-\,2^{2}\cdot 5\cdot 263$ $S_4$ (as 4T5) trivial $6.14722581704$
4.0.5440.2 $x^{4} - 2 x^{3} - 8 x + 26$ $2^{6}\cdot 5\cdot 17$ $D_{4}$ (as 4T3) $[2]$ $2.89946869917$
4.2.5595.1 $x^{4} - x^{3} + 4 x^{2} - 1$ $-\,3\cdot 5\cdot 373$ $S_4$ (as 4T5) trivial $3.06939590519$
4.0.5620.1 $x^{4} - x^{3} + 2 x^{2} + 2 x + 2$ $2^{2}\cdot 5\cdot 281$ $S_4$ (as 4T5) trivial $6.22390670009$
4.2.5755.1 $x^{4} - x^{3} + 4 x + 1$ $-\,5\cdot 1151$ $S_4$ (as 4T5) trivial $2.25309819598$
4.0.5780.1 $x^{4} - x^{3} + 5 x^{2} - 5 x + 2$ $2^{2}\cdot 5\cdot 17^{2}$ $S_4$ (as 4T5) trivial $5.39410731967$
4.2.6480.1 $x^{4} - 8 x + 3$ $-\,2^{4}\cdot 3^{4}\cdot 5$ $S_4$ (as 4T5) trivial $4.83550294476$
4.0.6480.1 $x^{4} - 2 x + 3$ $2^{4}\cdot 3^{4}\cdot 5$ $S_4$ (as 4T5) trivial $5.23767114534$
4.0.6685.1 $x^{4} - 2 x^{3} + 4 x^{2} - x + 3$ $5\cdot 7\cdot 191$ $S_4$ (as 4T5) trivial $2.49243428852$
4.2.6880.1 $x^{4} - 2 x^{3} - x^{2} - 4 x + 2$ $-\,2^{5}\cdot 5\cdot 43$ $S_4$ (as 4T5) trivial $7.31311030465$
4.0.6885.1 $x^{4} - x + 3$ $3^{4}\cdot 5\cdot 17$ $S_4$ (as 4T5) trivial $3.26832833556$
4.0.7065.1 $x^{4} - x^{2} - 3 x + 4$ $3^{2}\cdot 5\cdot 157$ $S_4$ (as 4T5) $[2]$ $1.79922408268$
4.2.7155.1 $x^{4} - x^{3} - 3$ $-\,3^{3}\cdot 5\cdot 53$ $S_4$ (as 4T5) trivial $4.71614770996$
4.2.7335.1 $x^{4} - 2 x^{3} - x^{2} - x - 2$ $-\,3^{2}\cdot 5\cdot 163$ $S_4$ (as 4T5) trivial $4.94493333958$
4.0.7540.1 $x^{4} - x^{3} + 5 x^{2} - x + 6$ $2^{2}\cdot 5\cdot 13\cdot 29$ $S_4$ (as 4T5) trivial $7.58827081214$
4.0.8080.1 $x^{4} - 2 x^{3} - 9 x^{2} + 10 x + 29$ $2^{4}\cdot 5\cdot 101$ $D_{4}$ (as 4T3) trivial $4.70656614512$
4.0.8080.2 $x^{4} - 9 x^{2} - 6 x + 34$ $2^{4}\cdot 5\cdot 101$ $D_{4}$ (as 4T3) trivial $12.5102664247$
4.2.8180.1 $x^{4} - x^{3} - 5 x - 3$ $-\,2^{2}\cdot 5\cdot 409$ $S_4$ (as 4T5) trivial $6.93299717372$
4.0.8285.1 $x^{4} - x^{3} + x^{2} - 5 x + 5$ $5\cdot 1657$ $S_4$ (as 4T5) trivial $1.70257522914$
4.0.8545.1 $x^{4} - 2 x^{3} + 3 x^{2} + x + 2$ $5\cdot 1709$ $S_4$ (as 4T5) trivial $4.48681235091$
4.2.8640.2 $x^{4} - 2 x^{3} - 3 x^{2} + 4 x - 2$ $-\,2^{6}\cdot 3^{3}\cdot 5$ $D_{4}$ (as 4T3) trivial $9.96319649523$
4.2.8640.3 $x^{4} - 2 x^{3} + 6 x^{2} - 6$ $-\,2^{6}\cdot 3^{3}\cdot 5$ $S_4$ (as 4T5) trivial $7.82566955574$
4.2.8960.1 $x^{4} - 4 x - 2$ $-\,2^{8}\cdot 5\cdot 7$ $S_4$ (as 4T5) trivial $6.77162446352$
4.2.9015.1 $x^{4} - 5 x^{2} - 3 x + 3$ $-\,3\cdot 5\cdot 601$ $S_4$ (as 4T5) trivial $7.01217458565$
4.2.9260.1 $x^{4} - 2 x^{3} + 4 x^{2} - 2 x - 4$ $-\,2^{2}\cdot 5\cdot 463$ $S_4$ (as 4T5) trivial $10.4122675289$
4.0.9360.1 $x^{4} - 2 x^{3} - 9 x^{2} + 10 x + 34$ $2^{4}\cdot 3^{2}\cdot 5\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $6.65026233206$
4.2.9395.1 $x^{4} - x^{3} - 2 x^{2} - 2 x + 3$ $-\,5\cdot 1879$ $S_4$ (as 4T5) trivial $3.95160638567$
4.2.9680.1 $x^{4} + x^{2} - 4 x - 1$ $-\,2^{4}\cdot 5\cdot 11^{2}$ $S_4$ (as 4T5) trivial $5.01336844406$
4.0.9680.1 $x^{4} + 3 x^{2} + 5$ $2^{4}\cdot 5\cdot 11^{2}$ $D_{4}$ (as 4T3) $[2]$ $2.3394720706$
Next   To download results, determine the number of results.