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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 43890ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 43890.ct8 | 43890ct1 | \([1, 0, 0, 4224, 17842176]\) | \(8334681620170751/137523678664458240\) | \(-137523678664458240\) | \([6]\) | \(497664\) | \(1.9674\) | \(\Gamma_0(N)\)-optimal |
| 43890.ct6 | 43890ct2 | \([1, 0, 0, -999296, 377303040]\) | \(110358600993178429667329/2339305154932838400\) | \(2339305154932838400\) | \([2, 6]\) | \(995328\) | \(2.3140\) | |
| 43890.ct7 | 43890ct3 | \([1, 0, 0, -38016, -481747200]\) | \(-6076082794014148609/100253882690711904000\) | \(-100253882690711904000\) | \([2]\) | \(1492992\) | \(2.5168\) | |
| 43890.ct5 | 43890ct4 | \([1, 0, 0, -2148416, -652538304]\) | \(1096677312076899338462209/450803852032204440000\) | \(450803852032204440000\) | \([6]\) | \(1990656\) | \(2.6606\) | |
| 43890.ct3 | 43890ct5 | \([1, 0, 0, -15906496, 24416653760]\) | \(445089424735238304524848129/206488340640267840\) | \(206488340640267840\) | \([6]\) | \(1990656\) | \(2.6606\) | |
| 43890.ct4 | 43890ct6 | \([1, 0, 0, -9449936, -11024979984]\) | \(93327647066813251630073089/1506876757438610250000\) | \(1506876757438610250000\) | \([2, 2]\) | \(2985984\) | \(2.8633\) | |
| 43890.ct1 | 43890ct7 | \([1, 0, 0, -150608156, -711423835980]\) | \(377806291534052689568887263169/100912963819335937500\) | \(100912963819335937500\) | \([2]\) | \(5971968\) | \(3.2099\) | |
| 43890.ct2 | 43890ct8 | \([1, 0, 0, -18882436, 14644625516]\) | \(744556702832013561199553089/338208906180283330846500\) | \(338208906180283330846500\) | \([2]\) | \(5971968\) | \(3.2099\) |
Rank
sage: E.rank()
The elliptic curves in class 43890ct have rank \(0\).
Complex multiplication
The elliptic curves in class 43890ct do not have complex multiplication.Modular form 43890.2.a.ct
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
