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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4400.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4400.e1 | 4400t4 | \([0, 1, 0, -177508, 28726488]\) | \(154639330142416/33275\) | \(133100000000\) | \([2]\) | \(20736\) | \(1.5192\) | |
4400.e2 | 4400t3 | \([0, 1, 0, -11133, 442738]\) | \(610462990336/8857805\) | \(2214451250000\) | \([2]\) | \(10368\) | \(1.1726\) | |
4400.e3 | 4400t2 | \([0, 1, 0, -2508, 26488]\) | \(436334416/171875\) | \(687500000000\) | \([2]\) | \(6912\) | \(0.96986\) | |
4400.e4 | 4400t1 | \([0, 1, 0, -1133, -14762]\) | \(643956736/15125\) | \(3781250000\) | \([2]\) | \(3456\) | \(0.62329\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4400.e have rank \(1\).
Complex multiplication
The elliptic curves in class 4400.e do not have complex multiplication.Modular form 4400.2.a.e
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.