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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 112112bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112112.bb4 | 112112bf1 | \([0, 0, 0, -12299, -4426758]\) | \(-426957777/17320303\) | \(-8346486078042112\) | \([2]\) | \(466944\) | \(1.7352\) | \(\Gamma_0(N)\)-optimal |
| 112112.bb3 | 112112bf2 | \([0, 0, 0, -486619, -129931830]\) | \(26444947540257/169338169\) | \(81602421738213376\) | \([2, 2]\) | \(933888\) | \(2.0818\) | |
| 112112.bb2 | 112112bf3 | \([0, 0, 0, -788459, 50508122]\) | \(112489728522417/62811265517\) | \(30268139834611847168\) | \([4]\) | \(1867776\) | \(2.4283\) | |
| 112112.bb1 | 112112bf4 | \([0, 0, 0, -7773899, -8342696390]\) | \(107818231938348177/4463459\) | \(2150897614401536\) | \([2]\) | \(1867776\) | \(2.4283\) |
Rank
sage: E.rank()
The elliptic curves in class 112112bf have rank \(0\).
Complex multiplication
The elliptic curves in class 112112bf do not have complex multiplication.Modular form 112112.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
