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SageMath
E = EllipticCurve("hi1")
E.isogeny_class()
Elliptic curves in class 248430hi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.hi7 | 248430hi1 | \([1, 1, 1, -4119970, -3219245905]\) | \(13619385906841/6048000\) | \(3434473236343968000\) | \([2]\) | \(10616832\) | \(2.5152\) | \(\Gamma_0(N)\)-optimal |
| 248430.hi6 | 248430hi2 | \([1, 1, 1, -4782450, -2115289233]\) | \(21302308926361/8930250000\) | \(5071214388039140250000\) | \([2, 2]\) | \(21233664\) | \(2.8618\) | |
| 248430.hi5 | 248430hi3 | \([1, 1, 1, -12193945, 12450724775]\) | \(353108405631241/86318776320\) | \(49017778945932782469120\) | \([2]\) | \(31850496\) | \(3.0645\) | |
| 248430.hi4 | 248430hi4 | \([1, 1, 1, -36084630, 81949845375]\) | \(9150443179640281/184570312500\) | \(104811805308348632812500\) | \([2]\) | \(42467328\) | \(3.2084\) | |
| 248430.hi8 | 248430hi5 | \([1, 1, 1, 15920050, -15513947233]\) | \(785793873833639/637994920500\) | \(-362297698310292257740500\) | \([2]\) | \(42467328\) | \(3.2084\) | |
| 248430.hi2 | 248430hi6 | \([1, 1, 1, -181788825, 943255264167]\) | \(1169975873419524361/108425318400\) | \(61571404462115274854400\) | \([2, 2]\) | \(63700992\) | \(3.4111\) | |
| 248430.hi1 | 248430hi7 | \([1, 1, 1, -2908556505, 60374792910375]\) | \(4791901410190533590281/41160000\) | \(23373498414007560000\) | \([2]\) | \(127401984\) | \(3.7577\) | |
| 248430.hi3 | 248430hi8 | \([1, 1, 1, -168539225, 1086600036647]\) | \(-932348627918877961/358766164249920\) | \(-203732273350220624194086720\) | \([2]\) | \(127401984\) | \(3.7577\) |
Rank
sage: E.rank()
The elliptic curves in class 248430hi have rank \(0\).
Complex multiplication
The elliptic curves in class 248430hi do not have complex multiplication.Modular form 248430.2.a.hi
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
