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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 76230dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.dc5 | 76230dl1 | \([1, -1, 1, -362473673, 3112393469001]\) | \(-4078208988807294650401/880065599546327040\) | \(-1136576532432862303758581760\) | \([2]\) | \(44236800\) | \(3.9125\) | \(\Gamma_0(N)\)-optimal |
| 76230.dc4 | 76230dl2 | \([1, -1, 1, -6071969993, 182109670698057]\) | \(19170300594578891358373921/671785075055001600\) | \(867588906485795479643750400\) | \([2, 2]\) | \(88473600\) | \(4.2591\) | |
| 76230.dc3 | 76230dl3 | \([1, -1, 1, -6345178313, 164824654558281]\) | \(21876183941534093095979041/3572502915711058560000\) | \(4613773084799938989323264640000\) | \([2, 2]\) | \(176947200\) | \(4.6057\) | |
| 76230.dc1 | 76230dl4 | \([1, -1, 1, -97150702793, 11655151915541577]\) | \(78519570041710065450485106721/96428056919040\) | \(124533746823848986229760\) | \([2]\) | \(176947200\) | \(4.6057\) | |
| 76230.dc6 | 76230dl5 | \([1, -1, 1, 11512156567, 924997114531977]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-467275000037955342892754287500000\) | \([2]\) | \(353894400\) | \(4.9523\) | |
| 76230.dc2 | 76230dl6 | \([1, -1, 1, -28573846313, -1701592116860919]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(242260247806592453120616783141600\) | \([2]\) | \(353894400\) | \(4.9523\) |
Rank
sage: E.rank()
The elliptic curves in class 76230dl have rank \(0\).
Complex multiplication
The elliptic curves in class 76230dl do not have complex multiplication.Modular form 76230.2.a.dl
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
