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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 75690.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75690.m1 | 75690t7 | \([1, -1, 0, -40369419, 98735028925]\) | \(16778985534208729/81000\) | \(35123722281729000\) | \([2]\) | \(4644864\) | \(2.7971\) | |
| 75690.m2 | 75690t8 | \([1, -1, 0, -3432699, 334032493]\) | \(10316097499609/5859375000\) | \(2540778521537109375000\) | \([2]\) | \(4644864\) | \(2.7971\) | |
| 75690.m3 | 75690t6 | \([1, -1, 0, -2524419, 1541499925]\) | \(4102915888729/9000000\) | \(3902635809081000000\) | \([2, 2]\) | \(2322432\) | \(2.4505\) | |
| 75690.m4 | 75690t5 | \([1, -1, 0, -2183814, -1241583530]\) | \(2656166199049/33750\) | \(14634884284053750\) | \([2]\) | \(1548288\) | \(2.2478\) | |
| 75690.m5 | 75690t4 | \([1, -1, 0, -518634, 123954898]\) | \(35578826569/5314410\) | \(2304467418904239690\) | \([2]\) | \(1548288\) | \(2.2478\) | |
| 75690.m6 | 75690t2 | \([1, -1, 0, -140184, -18266612]\) | \(702595369/72900\) | \(31611350053556100\) | \([2, 2]\) | \(774144\) | \(1.9012\) | |
| 75690.m7 | 75690t3 | \([1, -1, 0, -102339, 41263573]\) | \(-273359449/1536000\) | \(-666049844749824000\) | \([2]\) | \(1161216\) | \(2.1040\) | |
| 75690.m8 | 75690t1 | \([1, -1, 0, 11196, -1402880]\) | \(357911/2160\) | \(-936632594179440\) | \([2]\) | \(387072\) | \(1.5547\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75690.m have rank \(1\).
Complex multiplication
The elliptic curves in class 75690.m do not have complex multiplication.Modular form 75690.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
