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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 18150m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18150.d8 | 18150m1 | \([1, 1, 0, 4475, -351875]\) | \(357911/2160\) | \(-59790183750000\) | \([2]\) | \(69120\) | \(1.3254\) | \(\Gamma_0(N)\)-optimal |
| 18150.d6 | 18150m2 | \([1, 1, 0, -56025, -4647375]\) | \(702595369/72900\) | \(2017918701562500\) | \([2, 2]\) | \(138240\) | \(1.6720\) | |
| 18150.d7 | 18150m3 | \([1, 1, 0, -40900, 10402000]\) | \(-273359449/1536000\) | \(-42517464000000000\) | \([2]\) | \(207360\) | \(1.8747\) | |
| 18150.d4 | 18150m4 | \([1, 1, 0, -872775, -314195625]\) | \(2656166199049/33750\) | \(934221621093750\) | \([2]\) | \(276480\) | \(2.0185\) | |
| 18150.d5 | 18150m5 | \([1, 1, 0, -207275, 31198875]\) | \(35578826569/5314410\) | \(147106273343906250\) | \([2]\) | \(276480\) | \(2.0185\) | |
| 18150.d3 | 18150m6 | \([1, 1, 0, -1008900, 388890000]\) | \(4102915888729/9000000\) | \(249125765625000000\) | \([2, 2]\) | \(414720\) | \(2.2213\) | |
| 18150.d2 | 18150m7 | \([1, 1, 0, -1371900, 83607000]\) | \(10316097499609/5859375000\) | \(162191253662109375000\) | \([2]\) | \(829440\) | \(2.5678\) | |
| 18150.d1 | 18150m8 | \([1, 1, 0, -16133900, 24936765000]\) | \(16778985534208729/81000\) | \(2242131890625000\) | \([2]\) | \(829440\) | \(2.5678\) |
Rank
sage: E.rank()
The elliptic curves in class 18150m have rank \(0\).
Complex multiplication
The elliptic curves in class 18150m do not have complex multiplication.Modular form 18150.2.a.m
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.
