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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 25200.ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.ei1 | 25200ef8 | \([0, 0, 0, -23223675, 27101214250]\) | \(29689921233686449/10380965400750\) | \(484334321737392000000000\) | \([4]\) | \(2654208\) | \(3.2462\) | |
| 25200.ei2 | 25200ef5 | \([0, 0, 0, -20739675, 36353898250]\) | \(21145699168383889/2593080\) | \(120982740480000000\) | \([4]\) | \(884736\) | \(2.6969\) | |
| 25200.ei3 | 25200ef6 | \([0, 0, 0, -9723675, -11360285750]\) | \(2179252305146449/66177562500\) | \(3087580356000000000000\) | \([2, 2]\) | \(1327104\) | \(2.8996\) | |
| 25200.ei4 | 25200ef3 | \([0, 0, 0, -9651675, -11541221750]\) | \(2131200347946769/2058000\) | \(96018048000000000\) | \([2]\) | \(663552\) | \(2.5531\) | |
| 25200.ei5 | 25200ef2 | \([0, 0, 0, -1299675, 564858250]\) | \(5203798902289/57153600\) | \(2666558361600000000\) | \([2, 2]\) | \(442368\) | \(2.3503\) | |
| 25200.ei6 | 25200ef4 | \([0, 0, 0, -291675, 1418634250]\) | \(-58818484369/18600435000\) | \(-867821895360000000000\) | \([2]\) | \(884736\) | \(2.6969\) | |
| 25200.ei7 | 25200ef1 | \([0, 0, 0, -147675, -7685750]\) | \(7633736209/3870720\) | \(180592312320000000\) | \([2]\) | \(221184\) | \(2.0037\) | \(\Gamma_0(N)\)-optimal |
| 25200.ei8 | 25200ef7 | \([0, 0, 0, 2624325, -38241881750]\) | \(42841933504271/13565917968750\) | \(-632931468750000000000000\) | \([2]\) | \(2654208\) | \(3.2462\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.ei have rank \(0\).
Complex multiplication
The elliptic curves in class 25200.ei do not have complex multiplication.Modular form 25200.2.a.ei
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
