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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 30800ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30800.u3 | 30800ca1 | \([0, 1, 0, -22408, -25132812]\) | \(-19443408769/4249907200\) | \(-271994060800000000\) | \([2]\) | \(331776\) | \(2.0247\) | \(\Gamma_0(N)\)-optimal |
| 30800.u2 | 30800ca2 | \([0, 1, 0, -1430408, -653100812]\) | \(5057359576472449/51765560000\) | \(3312995840000000000\) | \([2]\) | \(663552\) | \(2.3712\) | |
| 30800.u4 | 30800ca3 | \([0, 1, 0, 201592, 676883188]\) | \(14156681599871/3100231750000\) | \(-198414832000000000000\) | \([2]\) | \(995328\) | \(2.5740\) | |
| 30800.u1 | 30800ca4 | \([0, 1, 0, -10446408, 12623939188]\) | \(1969902499564819009/63690429687500\) | \(4076187500000000000000\) | \([2]\) | \(1990656\) | \(2.9205\) |
Rank
sage: E.rank()
The elliptic curves in class 30800ca have rank \(0\).
Complex multiplication
The elliptic curves in class 30800ca do not have complex multiplication.Modular form 30800.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
