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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 6720cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.bq7 | 6720cd1 | \([0, 1, 0, 13439, -447361]\) | \(1023887723039/928972800\) | \(-243524645683200\) | \([2]\) | \(24576\) | \(1.4479\) | \(\Gamma_0(N)\)-optimal |
| 6720.bq6 | 6720cd2 | \([0, 1, 0, -68481, -4068225]\) | \(135487869158881/51438240000\) | \(13484225986560000\) | \([2, 2]\) | \(49152\) | \(1.7944\) | |
| 6720.bq4 | 6720cd3 | \([0, 1, 0, -964481, -364797825]\) | \(378499465220294881/120530818800\) | \(31596430963507200\) | \([2]\) | \(98304\) | \(2.1410\) | |
| 6720.bq5 | 6720cd4 | \([0, 1, 0, -483201, 126236799]\) | \(47595748626367201/1215506250000\) | \(318637670400000000\) | \([2, 2]\) | \(98304\) | \(2.1410\) | |
| 6720.bq2 | 6720cd5 | \([0, 1, 0, -7683201, 8194556799]\) | \(191342053882402567201/129708022500\) | \(34002179850240000\) | \([2, 2]\) | \(196608\) | \(2.4876\) | |
| 6720.bq8 | 6720cd6 | \([0, 1, 0, 81279, 404073855]\) | \(226523624554079/269165039062500\) | \(-70560000000000000000\) | \([2]\) | \(196608\) | \(2.4876\) | |
| 6720.bq1 | 6720cd7 | \([0, 1, 0, -122931201, 524574745599]\) | \(783736670177727068275201/360150\) | \(94411161600\) | \([2]\) | \(393216\) | \(2.8341\) | |
| 6720.bq3 | 6720cd8 | \([0, 1, 0, -7635201, 8302047999]\) | \(-187778242790732059201/4984939585440150\) | \(-1306772002685622681600\) | \([2]\) | \(393216\) | \(2.8341\) |
Rank
sage: E.rank()
The elliptic curves in class 6720cd have rank \(0\).
Complex multiplication
The elliptic curves in class 6720cd do not have complex multiplication.Modular form 6720.2.a.cd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
