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SageMath
E = EllipticCurve("il1")
E.isogeny_class()
Elliptic curves in class 394944.il
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
394944.il1 | 394944il5 | \([0, 1, 0, -3058446497, 65101747163583]\) | \(6812873765474836663297/74052\) | \(34390051434528768\) | \([2]\) | \(94371840\) | \(3.6766\) | |
394944.il2 | 394944il3 | \([0, 1, 0, -191153057, 1017165320895]\) | \(1663303207415737537/5483698704\) | \(2546652088829724327936\) | \([2, 2]\) | \(47185920\) | \(3.3300\) | |
394944.il3 | 394944il6 | \([0, 1, 0, -188520097, 1046549681087]\) | \(-1595514095015181697/95635786040388\) | \(-44413649879956404481032192\) | \([2]\) | \(94371840\) | \(3.6766\) | |
394944.il4 | 394944il2 | \([0, 1, 0, -12111777, 15429359295]\) | \(423108074414017/23284318464\) | \(10813332652661749579776\) | \([2, 2]\) | \(23592960\) | \(2.9835\) | |
394944.il5 | 394944il1 | \([0, 1, 0, -2199457, -951740737]\) | \(2533811507137/625016832\) | \(290260371089588748288\) | \([2]\) | \(11796480\) | \(2.6369\) | \(\Gamma_0(N)\)-optimal |
394944.il6 | 394944il4 | \([0, 1, 0, 8332383, 62242396863]\) | \(137763859017023/3683199928848\) | \(-1710493099399549217144832\) | \([2]\) | \(47185920\) | \(3.3300\) |
Rank
sage: E.rank()
The elliptic curves in class 394944.il have rank \(1\).
Complex multiplication
The elliptic curves in class 394944.il do not have complex multiplication.Modular form 394944.2.a.il
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.