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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 185130fr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.bg3 | 185130fr1 | \([1, -1, 0, -43613565, -110856046779]\) | \(-191808834096148160787/11043434659840\) | \(-528231190034361876480\) | \([2]\) | \(19353600\) | \(3.0398\) | \(\Gamma_0(N)\)-optimal |
| 185130.bg2 | 185130fr2 | \([1, -1, 0, -697826685, -7095104473275]\) | \(785681552361835673854227/2604236800\) | \(124566237440409600\) | \([2]\) | \(38707200\) | \(3.3863\) | |
| 185130.bg4 | 185130fr3 | \([1, -1, 0, -4583805, -299896255675]\) | \(-305460292990923/1114070936704000\) | \(-38847247108570145722752000\) | \([2]\) | \(58060800\) | \(3.5891\) | |
| 185130.bg1 | 185130fr4 | \([1, -1, 0, -700324125, -7041759104539]\) | \(1089365384367428097483/16063552169500000\) | \(560130203572282136128500000\) | \([2]\) | \(116121600\) | \(3.9357\) |
Rank
sage: E.rank()
The elliptic curves in class 185130fr have rank \(1\).
Complex multiplication
The elliptic curves in class 185130fr do not have complex multiplication.Modular form 185130.2.a.fr
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.
