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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 60840ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
60840.bb4 | 60840ca1 | \([0, 0, 0, -30927, -7430254]\) | \(-3631696/24375\) | \(-21956961068640000\) | \([4]\) | \(516096\) | \(1.8190\) | \(\Gamma_0(N)\)-optimal |
60840.bb3 | 60840ca2 | \([0, 0, 0, -791427, -270411154]\) | \(15214885924/38025\) | \(137011437068313600\) | \([2, 2]\) | \(1032192\) | \(2.1656\) | |
60840.bb2 | 60840ca3 | \([0, 0, 0, -1095627, -43417114]\) | \(20183398562/11567205\) | \(83357758312361994240\) | \([2]\) | \(2064384\) | \(2.5122\) | |
60840.bb1 | 60840ca4 | \([0, 0, 0, -12655227, -17328182794]\) | \(31103978031362/195\) | \(1405245508392960\) | \([2]\) | \(2064384\) | \(2.5122\) |
Rank
sage: E.rank()
The elliptic curves in class 60840ca have rank \(0\).
Complex multiplication
The elliptic curves in class 60840ca do not have complex multiplication.Modular form 60840.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.