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SageMath
E = EllipticCurve("vc1")
E.isogeny_class()
Elliptic curves in class 235200.vc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235200.vc1 | 235200vc4 | \([0, 1, 0, -69189633, -221541127137]\) | \(608119035935048/826875\) | \(49807880640000000000\) | \([2]\) | \(14155776\) | \(3.0538\) | |
235200.vc2 | 235200vc3 | \([0, 1, 0, -10977633, 9348832863]\) | \(2428799546888/778248135\) | \(46878778795322880000000\) | \([2]\) | \(14155776\) | \(3.0538\) | |
235200.vc3 | 235200vc2 | \([0, 1, 0, -4362633, -3398272137]\) | \(1219555693504/43758225\) | \(329479130433600000000\) | \([2, 2]\) | \(7077888\) | \(2.7072\) | |
235200.vc4 | 235200vc1 | \([0, 1, 0, 102492, -187847262]\) | \(1012048064/130203045\) | \(-15318258041205000000\) | \([2]\) | \(3538944\) | \(2.3606\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235200.vc have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.vc do not have complex multiplication.Modular form 235200.2.a.vc
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.