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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 20400cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20400.dc5 | 20400cx1 | \([0, 1, 0, -13608, -571212]\) | \(4354703137/352512\) | \(22560768000000\) | \([2]\) | \(49152\) | \(1.3052\) | \(\Gamma_0(N)\)-optimal |
| 20400.dc4 | 20400cx2 | \([0, 1, 0, -45608, 3076788]\) | \(163936758817/30338064\) | \(1941636096000000\) | \([2, 2]\) | \(98304\) | \(1.6518\) | |
| 20400.dc2 | 20400cx3 | \([0, 1, 0, -693608, 222100788]\) | \(576615941610337/27060804\) | \(1731891456000000\) | \([2, 2]\) | \(196608\) | \(1.9984\) | |
| 20400.dc6 | 20400cx4 | \([0, 1, 0, 90392, 18036788]\) | \(1276229915423/2927177028\) | \(-187339329792000000\) | \([2]\) | \(196608\) | \(1.9984\) | |
| 20400.dc1 | 20400cx5 | \([0, 1, 0, -11097608, 14225884788]\) | \(2361739090258884097/5202\) | \(332928000000\) | \([2]\) | \(393216\) | \(2.3449\) | |
| 20400.dc3 | 20400cx6 | \([0, 1, 0, -657608, 246220788]\) | \(-491411892194497/125563633938\) | \(-8036072572032000000\) | \([2]\) | \(393216\) | \(2.3449\) |
Rank
sage: E.rank()
The elliptic curves in class 20400cx have rank \(1\).
Complex multiplication
The elliptic curves in class 20400cx do not have complex multiplication.Modular form 20400.2.a.cx
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
