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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 38640.v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38640.v1 | 38640br4 | \([0, -1, 0, -6924496, -7011120704]\) | \(8964546681033941529169/31696875000\) | \(129830400000000\) | \([2]\) | \(884736\) | \(2.3504\) | |
38640.v2 | 38640br3 | \([0, -1, 0, -576976, -30194240]\) | \(5186062692284555089/2903809817953800\) | \(11894005014338764800\) | \([4]\) | \(884736\) | \(2.3504\) | |
38640.v3 | 38640br2 | \([0, -1, 0, -432976, -109336640]\) | \(2191574502231419089/4115217960000\) | \(16855932764160000\) | \([2, 2]\) | \(442368\) | \(2.0039\) | |
38640.v4 | 38640br1 | \([0, -1, 0, -18256, -2836544]\) | \(-164287467238609/757170892800\) | \(-3101371976908800\) | \([2]\) | \(221184\) | \(1.6573\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 38640.v have rank \(2\).
Complex multiplication
The elliptic curves in class 38640.v do not have complex multiplication.Modular form 38640.2.a.v
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.