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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 13475a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13475.n1 | 13475a1 | \([0, 1, 1, -109433, -13970456]\) | \(-78843215872/539\) | \(-990825171875\) | \([]\) | \(34560\) | \(1.4827\) | \(\Gamma_0(N)\)-optimal |
| 13475.n2 | 13475a2 | \([0, 1, 1, -60433, -26459331]\) | \(-13278380032/156590819\) | \(-287855519758296875\) | \([]\) | \(103680\) | \(2.0320\) | |
| 13475.n3 | 13475a3 | \([0, 1, 1, 539817, 684536794]\) | \(9463555063808/115539436859\) | \(-212392175109757671875\) | \([]\) | \(311040\) | \(2.5813\) |
Rank
sage: E.rank()
The elliptic curves in class 13475a have rank \(0\).
Complex multiplication
The elliptic curves in class 13475a do not have complex multiplication.Modular form 13475.2.a.a
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.
