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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 190740.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190740.k1 | 190740u4 | \([0, -1, 0, -72248940, -236347539288]\) | \(6749703004355978704/5671875\) | \(35047750188000000\) | \([2]\) | \(8957952\) | \(2.9102\) | |
| 190740.k2 | 190740u3 | \([0, -1, 0, -4514565, -3693508038]\) | \(-26348629355659264/24169921875\) | \(-9334450511718750000\) | \([2]\) | \(4478976\) | \(2.5636\) | |
| 190740.k3 | 190740u2 | \([0, -1, 0, -912180, -308467800]\) | \(13584145739344/1195803675\) | \(7389131191236115200\) | \([2]\) | \(2985984\) | \(2.3609\) | |
| 190740.k4 | 190740u1 | \([0, -1, 0, 63195, -22487850]\) | \(72268906496/606436875\) | \(-234206590631310000\) | \([2]\) | \(1492992\) | \(2.0143\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 190740.k have rank \(1\).
Complex multiplication
The elliptic curves in class 190740.k do not have complex multiplication.Modular form 190740.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
