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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 307230.gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 307230.gd1 | 307230gd7 | \([1, 1, 1, -7379799645, 244010995941495]\) | \(377806291534052689568887263169/100912963819335937500\) | \(11872309280381053710937500\) | \([4]\) | \(286654464\) | \(4.1829\) | |
| 307230.gd2 | 307230gd8 | \([1, 1, 1, -925239365, -5024031791353]\) | \(744556702832013561199553089/338208906180283330846500\) | \(39789939603204153590759878500\) | \([2]\) | \(286654464\) | \(4.1829\) | |
| 307230.gd3 | 307230gd5 | \([1, 1, 1, -779418305, -8375691657985]\) | \(445089424735238304524848129/206488340640267840\) | \(24293146787986871108160\) | \([2]\) | \(95551488\) | \(3.6336\) | |
| 307230.gd4 | 307230gd6 | \([1, 1, 1, -463046865, 3781105087647]\) | \(93327647066813251630073089/1506876757438610250000\) | \(177282543635895057302250000\) | \([2, 2]\) | \(143327232\) | \(3.8363\) | |
| 307230.gd5 | 307230gd4 | \([1, 1, 1, -105272385, 223715365887]\) | \(1096677312076899338462209/450803852032204440000\) | \(53036622387736820161560000\) | \([4]\) | \(95551488\) | \(3.6336\) | |
| 307230.gd6 | 307230gd2 | \([1, 1, 1, -48965505, -129463908225]\) | \(110358600993178429667329/2339305154932838400\) | \(275216912172693504921600\) | \([2, 2]\) | \(47775744\) | \(3.2870\) | |
| 307230.gd7 | 307230gd3 | \([1, 1, 1, -1862785, 165237426815]\) | \(-6076082794014148609/100253882690711904000\) | \(-11794769044679564793696000\) | \([2]\) | \(71663616\) | \(3.4897\) | |
| 307230.gd8 | 307230gd1 | \([1, 1, 1, 206975, -6119659393]\) | \(8334681620170751/137523678664458240\) | \(-16179523271194847477760\) | \([2]\) | \(23887872\) | \(2.9404\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 307230.gd have rank \(1\).
Complex multiplication
The elliptic curves in class 307230.gd do not have complex multiplication.Modular form 307230.2.a.gd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 4 & 12 \\ 4 & 1 & 3 & 2 & 12 & 6 & 4 & 12 \\ 12 & 3 & 1 & 6 & 4 & 2 & 12 & 4 \\ 2 & 2 & 6 & 1 & 6 & 3 & 2 & 6 \\ 3 & 12 & 4 & 6 & 1 & 2 & 12 & 4 \\ 6 & 6 & 2 & 3 & 2 & 1 & 6 & 2 \\ 4 & 4 & 12 & 2 & 12 & 6 & 1 & 3 \\ 12 & 12 & 4 & 6 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
