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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 15680.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15680.cm1 | 15680j3 | \([0, 1, 0, -25741, 1654309]\) | \(-250523582464/13671875\) | \(-102942875000000\) | \([]\) | \(41472\) | \(1.4470\) | |
15680.cm2 | 15680j1 | \([0, 1, 0, -261, -1891]\) | \(-262144/35\) | \(-263533760\) | \([]\) | \(4608\) | \(0.34838\) | \(\Gamma_0(N)\)-optimal |
15680.cm3 | 15680j2 | \([0, 1, 0, 1699, 5165]\) | \(71991296/42875\) | \(-322828856000\) | \([]\) | \(13824\) | \(0.89769\) |
Rank
sage: E.rank()
The elliptic curves in class 15680.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 15680.cm do not have complex multiplication.Modular form 15680.2.a.cm
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.