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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 57330cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57330.bu6 | 57330cm1 | \([1, -1, 0, 6606, 81108]\) | \(371694959/249600\) | \(-21407223801600\) | \([2]\) | \(196608\) | \(1.2466\) | \(\Gamma_0(N)\)-optimal |
| 57330.bu5 | 57330cm2 | \([1, -1, 0, -28674, 694980]\) | \(30400540561/15210000\) | \(1304502700410000\) | \([2, 2]\) | \(393216\) | \(1.5932\) | |
| 57330.bu3 | 57330cm3 | \([1, -1, 0, -249174, -47329920]\) | \(19948814692561/231344100\) | \(19841486073236100\) | \([2, 2]\) | \(786432\) | \(1.9398\) | |
| 57330.bu2 | 57330cm4 | \([1, -1, 0, -372654, 87584328]\) | \(66730743078481/60937500\) | \(5226372998437500\) | \([2]\) | \(786432\) | \(1.9398\) | |
| 57330.bu4 | 57330cm5 | \([1, -1, 0, -50724, -120796110]\) | \(-168288035761/73415764890\) | \(-6296585374863291690\) | \([2]\) | \(1572864\) | \(2.2863\) | |
| 57330.bu1 | 57330cm6 | \([1, -1, 0, -3975624, -3050103330]\) | \(81025909800741361/11088090\) | \(950982468598890\) | \([2]\) | \(1572864\) | \(2.2863\) |
Rank
sage: E.rank()
The elliptic curves in class 57330cm have rank \(2\).
Complex multiplication
The elliptic curves in class 57330cm do not have complex multiplication.Modular form 57330.2.a.cm
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.
