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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 316680.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 316680.ca1 | 316680ca6 | \([0, 1, 0, -73804512080, -7717450778016672]\) | \(21709134044926784078766515550491042/675700487109375\) | \(1383834597600000000\) | \([2]\) | \(415236096\) | \(4.4026\) | |
| 316680.ca2 | 316680ca4 | \([0, 1, 0, -4612782200, -120586310880000]\) | \(10600164454357575717381622879204/1870115423354251880625\) | \(1914998193514753925760000\) | \([2, 2]\) | \(207618048\) | \(4.0561\) | |
| 316680.ca3 | 316680ca5 | \([0, 1, 0, -4598079200, -121393199757600]\) | \(-5249562401418013365341039985602/70420471124711699773447575\) | \(-144221124863409561136020633600\) | \([2]\) | \(415236096\) | \(4.4026\) | |
| 316680.ca4 | 316680ca3 | \([0, 1, 0, -548834720, 2004150739728]\) | \(17854532742609155739290901124/8637663210066342291347235\) | \(8844967127107934506339568640\) | \([4]\) | \(207618048\) | \(4.0561\) | |
| 316680.ca5 | 316680ca2 | \([0, 1, 0, -289218020, -1871615051232]\) | \(10451046837659932954397263696/137452321003663080549225\) | \(35187794176937748620601600\) | \([2, 4]\) | \(103809024\) | \(3.7095\) | |
| 316680.ca6 | 316680ca1 | \([0, 1, 0, -2769215, -77185157190]\) | \(-146782375668163487266816/160765018398963336090915\) | \(-2572240294383413377454640\) | \([4]\) | \(51904512\) | \(3.3629\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 316680.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 316680.ca do not have complex multiplication.Modular form 316680.2.a.ca
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
