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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 22320bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22320.k6 | 22320bn1 | \([0, 0, 0, 8637, -1750462]\) | \(23862997439/457113600\) | \(-1364933895782400\) | \([2]\) | \(98304\) | \(1.5846\) | \(\Gamma_0(N)\)-optimal |
| 22320.k5 | 22320bn2 | \([0, 0, 0, -175683, -26781118]\) | \(200828550012481/12454560000\) | \(37189116887040000\) | \([2, 2]\) | \(196608\) | \(1.9311\) | |
| 22320.k4 | 22320bn3 | \([0, 0, 0, -532803, 116709698]\) | \(5601911201812801/1271193750000\) | \(3795764198400000000\) | \([2]\) | \(393216\) | \(2.2777\) | |
| 22320.k2 | 22320bn4 | \([0, 0, 0, -2767683, -1772233918]\) | \(785209010066844481/3324675600\) | \(9927428146790400\) | \([2, 2]\) | \(393216\) | \(2.2777\) | |
| 22320.k3 | 22320bn5 | \([0, 0, 0, -2724483, -1830234238]\) | \(-749011598724977281/51173462246460\) | \(-152803139492533616640\) | \([2]\) | \(786432\) | \(2.6243\) | |
| 22320.k1 | 22320bn6 | \([0, 0, 0, -44282883, -113423212798]\) | \(3216206300355197383681/57660\) | \(172171837440\) | \([2]\) | \(786432\) | \(2.6243\) |
Rank
sage: E.rank()
The elliptic curves in class 22320bn have rank \(0\).
Complex multiplication
The elliptic curves in class 22320bn do not have complex multiplication.Modular form 22320.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
