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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 188760bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188760.ba4 | 188760bn1 | \([0, -1, 0, -11054116, -19252754684]\) | \(-329381898333928144/162600887109375\) | \(-73742691883103100000000\) | \([2]\) | \(20643840\) | \(3.0931\) | \(\Gamma_0(N)\)-optimal |
| 188760.ba3 | 188760bn2 | \([0, -1, 0, -194066616, -1040389299684]\) | \(445574312599094932036/61129333175625\) | \(110893406832582042240000\) | \([2, 2]\) | \(41287680\) | \(3.4397\) | |
| 188760.ba2 | 188760bn3 | \([0, -1, 0, -211369616, -843820298484]\) | \(287849398425814280018/81784533026485575\) | \(296727120103288446428313600\) | \([2]\) | \(82575360\) | \(3.7863\) | |
| 188760.ba1 | 188760bn4 | \([0, -1, 0, -3104963616, -66592625380884]\) | \(912446049969377120252018/17177299425\) | \(62321937913144166400\) | \([2]\) | \(82575360\) | \(3.7863\) |
Rank
sage: E.rank()
The elliptic curves in class 188760bn have rank \(1\).
Complex multiplication
The elliptic curves in class 188760bn do not have complex multiplication.Modular form 188760.2.a.bn
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.
