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SageMath
E = EllipticCurve("ka1")
E.isogeny_class()
Elliptic curves in class 95550.ka
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 95550.ka1 | 95550jx7 | \([1, 0, 0, -2775456188, 56269848550992]\) | \(1286229821345376481036009/247265484375000000\) | \(454539640175537109375000000\) | \([2]\) | \(95551488\) | \(4.1156\) | |
| 95550.ka2 | 95550jx8 | \([1, 0, 0, -1220784188, -15902247529008]\) | \(109454124781830273937129/3914078300576808000\) | \(7195115593508763818625000000\) | \([2]\) | \(95551488\) | \(4.1156\) | |
| 95550.ka3 | 95550jx5 | \([1, 0, 0, -1210108313, -16202692247883]\) | \(106607603143751752938169/5290068420\) | \(9724550930384062500\) | \([2]\) | \(31850496\) | \(3.5663\) | |
| 95550.ka4 | 95550jx6 | \([1, 0, 0, -191784188, 682145470992]\) | \(424378956393532177129/136231857216000000\) | \(250430340150081000000000000\) | \([2, 2]\) | \(47775744\) | \(3.7690\) | |
| 95550.ka5 | 95550jx4 | \([1, 0, 0, -84235313, -192009524883]\) | \(35958207000163259449/12145729518877500\) | \(22327077065100296835937500\) | \([2]\) | \(31850496\) | \(3.5663\) | |
| 95550.ka6 | 95550jx2 | \([1, 0, 0, -75635813, -253143370383]\) | \(26031421522845051769/5797789779600\) | \(10657877652815006250000\) | \([2, 2]\) | \(15925248\) | \(3.2197\) | |
| 95550.ka7 | 95550jx1 | \([1, 0, 0, -4193813, -4882420383]\) | \(-4437543642183289/3033210136320\) | \(-5575845926998620000000\) | \([4]\) | \(7962624\) | \(2.8731\) | \(\Gamma_0(N)\)-optimal |
| 95550.ka8 | 95550jx3 | \([1, 0, 0, 34007812, 72732862992]\) | \(2366200373628880151/2612420149248000\) | \(-4802322158419968000000000\) | \([4]\) | \(23887872\) | \(3.4225\) |
Rank
sage: E.rank()
The elliptic curves in class 95550.ka have rank \(1\).
Complex multiplication
The elliptic curves in class 95550.ka do not have complex multiplication.Modular form 95550.2.a.ka
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
