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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 253575ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
253575.ef1 | 253575ef1 | \([1, -1, 0, -45744792, -116062790509]\) | \(292583028222603/8456021875\) | \(305960707396362744140625\) | \([2]\) | \(29859840\) | \(3.2844\) | \(\Gamma_0(N)\)-optimal |
253575.ef2 | 253575ef2 | \([1, -1, 0, 10978833, -384989496634]\) | \(4044759171237/1771943359375\) | \(-64113486425987091064453125\) | \([2]\) | \(59719680\) | \(3.6310\) |
Rank
sage: E.rank()
The elliptic curves in class 253575ef have rank \(0\).
Complex multiplication
The elliptic curves in class 253575ef do not have complex multiplication.Modular form 253575.2.a.ef
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.