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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 56784.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56784.e1 | 56784cc3 | \([0, -1, 0, -70456832, 227658891264]\) | \(-1956469094246217097/36641439744\) | \(-724423602705600086016\) | \([]\) | \(7838208\) | \(3.1271\) | |
56784.e2 | 56784cc2 | \([0, -1, 0, -328592, 694378944]\) | \(-198461344537/10417365504\) | \(-205957667106802630656\) | \([]\) | \(2612736\) | \(2.5778\) | |
56784.e3 | 56784cc1 | \([0, -1, 0, 36448, -25479936]\) | \(270840023/14329224\) | \(-283297494492020736\) | \([]\) | \(870912\) | \(2.0285\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 56784.e have rank \(1\).
Complex multiplication
The elliptic curves in class 56784.e do not have complex multiplication.Modular form 56784.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.