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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 112710f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112710.d4 | 112710f1 | \([1, 1, 0, 1155272, 18488128]\) | \(7064514799444439/4094064000000\) | \(-98820752290416000000\) | \([2]\) | \(3732480\) | \(2.5261\) | \(\Gamma_0(N)\)-optimal |
| 112710.d3 | 112710f2 | \([1, 1, 0, -4624728, 142180128]\) | \(453198971846635561/261896250564000\) | \(6321538818829838916000\) | \([2]\) | \(7464960\) | \(2.8727\) | |
| 112710.d2 | 112710f3 | \([1, 1, 0, -15426103, -25161987947]\) | \(-16818951115904497561/1592332281446400\) | \(-38435030314339899801600\) | \([2]\) | \(11197440\) | \(3.0754\) | |
| 112710.d1 | 112710f4 | \([1, 1, 0, -252174903, -1541443352427]\) | \(73474353581350183614361/576510977802240\) | \(13915573505959036354560\) | \([2]\) | \(22394880\) | \(3.4220\) |
Rank
sage: E.rank()
The elliptic curves in class 112710f have rank \(0\).
Complex multiplication
The elliptic curves in class 112710f do not have complex multiplication.Modular form 112710.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
