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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 270400ch
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270400.ch2 | 270400ch1 | \([0, 1, 0, -1411713, -816771617]\) | \(-9836106385/3407872\) | \(-107801131355000012800\) | \([]\) | \(11612160\) | \(2.5553\) | \(\Gamma_0(N)\)-optimal |
270400.ch1 | 270400ch2 | \([0, 1, 0, -122550913, -522224116257]\) | \(-6434774386429585/140608\) | \(-4447849413817139200\) | \([]\) | \(34836480\) | \(3.1046\) |
Rank
sage: E.rank()
The elliptic curves in class 270400ch have rank \(1\).
Complex multiplication
The elliptic curves in class 270400ch do not have complex multiplication.Modular form 270400.2.a.ch
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.