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SageMath
E = EllipticCurve("eo1")
E.isogeny_class()
Elliptic curves in class 224400.eo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 224400.eo1 | 224400bf7 | \([0, 1, 0, -80494593008, 8790161436683988]\) | \(901247067798311192691198986281/552431869440\) | \(35355639644160000000\) | \([2]\) | \(382205952\) | \(4.4587\) | |
| 224400.eo2 | 224400bf8 | \([0, 1, 0, -5064705008, 135406313483988]\) | \(224494757451893010998773801/6152490825146276160000\) | \(393759412809361674240000000000\) | \([2]\) | \(382205952\) | \(4.4587\) | |
| 224400.eo3 | 224400bf6 | \([0, 1, 0, -5030913008, 137344960523988]\) | \(220031146443748723000125481/172266701724057600\) | \(11025068910339686400000000\) | \([2, 2]\) | \(191102976\) | \(4.1121\) | |
| 224400.eo4 | 224400bf4 | \([0, 1, 0, -993963008, 12052377623988]\) | \(1696892787277117093383481/1440538624914939000\) | \(92194471994556096000000000\) | \([2]\) | \(127401984\) | \(3.9093\) | |
| 224400.eo5 | 224400bf5 | \([0, 1, 0, -650955008, -6324479016012]\) | \(476646772170172569823801/5862293314453125000\) | \(375186772125000000000000000\) | \([2]\) | \(127401984\) | \(3.9093\) | |
| 224400.eo6 | 224400bf3 | \([0, 1, 0, -312321008, 2176174091988]\) | \(-52643812360427830814761/1504091705903677440\) | \(-96261869177835356160000000\) | \([2]\) | \(95551488\) | \(3.7655\) | |
| 224400.eo7 | 224400bf2 | \([0, 1, 0, -75963008, 98181623988]\) | \(757443433548897303481/373234243041000000\) | \(23886991554624000000000000\) | \([2, 2]\) | \(63700992\) | \(3.5628\) | |
| 224400.eo8 | 224400bf1 | \([0, 1, 0, 17348992, 11774711988]\) | \(9023321954633914439/6156756739584000\) | \(-394032431333376000000000\) | \([2]\) | \(31850496\) | \(3.2162\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 224400.eo have rank \(0\).
Complex multiplication
The elliptic curves in class 224400.eo do not have complex multiplication.Modular form 224400.2.a.eo
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
