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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 296450.dh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 296450.dh1 | 296450dh4 | \([1, -1, 0, -3099264317, -66409613011659]\) | \(1010962818911303721/57392720\) | \(186905114096586751250000\) | \([2]\) | \(141557760\) | \(3.9320\) | |
| 296450.dh2 | 296450dh3 | \([1, -1, 0, -324492317, 530670552341]\) | \(1160306142246441/634128110000\) | \(2065101405742800027968750000\) | \([2]\) | \(141557760\) | \(3.9320\) | |
| 296450.dh3 | 296450dh2 | \([1, -1, 0, -194054317, -1033672381659]\) | \(248158561089321/1859334400\) | \(6055107828583636900000000\) | \([2, 2]\) | \(70778880\) | \(3.5854\) | |
| 296450.dh4 | 296450dh1 | \([1, -1, 0, -4326317, -36651741659]\) | \(-2749884201/176619520\) | \(-575179073884011520000000\) | \([2]\) | \(35389440\) | \(3.2389\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 296450.dh have rank \(1\).
Complex multiplication
The elliptic curves in class 296450.dh do not have complex multiplication.Modular form 296450.2.a.dh
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
