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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 13104by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13104.h3 | 13104by1 | \([0, 0, 0, 1941, 312986]\) | \(270840023/14329224\) | \(-42786833596416\) | \([]\) | \(41472\) | \(1.2953\) | \(\Gamma_0(N)\)-optimal |
| 13104.h2 | 13104by2 | \([0, 0, 0, -17499, -8532214]\) | \(-198461344537/10417365504\) | \(-31106086717095936\) | \([]\) | \(124416\) | \(1.8446\) | |
| 13104.h1 | 13104by3 | \([0, 0, 0, -3752139, -2797522054]\) | \(-1956469094246217097/36641439744\) | \(-109410752812548096\) | \([]\) | \(373248\) | \(2.3939\) |
Rank
sage: E.rank()
The elliptic curves in class 13104by have rank \(0\).
Complex multiplication
The elliptic curves in class 13104by do not have complex multiplication.Modular form 13104.2.a.by
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
