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SageMath
E = EllipticCurve("ia1")
E.isogeny_class()
Elliptic curves in class 235950.ia
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235950.ia1 | 235950ia5 | \([1, 0, 0, -27270438, 54811063242]\) | \(81025909800741361/11088090\) | \(306925434507656250\) | \([2]\) | \(15728640\) | \(2.7677\) | |
| 235950.ia2 | 235950ia4 | \([1, 0, 0, -2556188, -1572002508]\) | \(66730743078481/60937500\) | \(1686789038085937500\) | \([2]\) | \(7864320\) | \(2.4212\) | |
| 235950.ia3 | 235950ia3 | \([1, 0, 0, -1709188, 851264492]\) | \(19948814692561/231344100\) | \(6403752892814062500\) | \([2, 2]\) | \(7864320\) | \(2.4212\) | |
| 235950.ia4 | 235950ia6 | \([1, 0, 0, -347938, 2170315742]\) | \(-168288035761/73415764890\) | \(-2032195404129582656250\) | \([2]\) | \(15728640\) | \(2.7677\) | |
| 235950.ia5 | 235950ia2 | \([1, 0, 0, -196688, -12373008]\) | \(30400540561/15210000\) | \(421022543906250000\) | \([2, 2]\) | \(3932160\) | \(2.0746\) | |
| 235950.ia6 | 235950ia1 | \([1, 0, 0, 45312, -1483008]\) | \(371694959/249600\) | \(-6909087900000000\) | \([2]\) | \(1966080\) | \(1.7280\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235950.ia have rank \(1\).
Complex multiplication
The elliptic curves in class 235950.ia do not have complex multiplication.Modular form 235950.2.a.ia
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
