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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 323400.ij
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.ij1 | 323400ij5 | \([0, 1, 0, -37271458408, 2769559791400688]\) | \(1520949008089505953959842/278553515625\) | \(1048689361912500000000000\) | \([2]\) | \(339738624\) | \(4.4450\) | |
| 323400.ij2 | 323400ij3 | \([0, 1, 0, -2329705408, 43264455328688]\) | \(742879737792994384804/317817082130625\) | \(598253790329374410000000000\) | \([2, 2]\) | \(169869312\) | \(4.0984\) | |
| 323400.ij3 | 323400ij6 | \([0, 1, 0, -1965880408, 57235335328688]\) | \(-223180773010681046402/246754509479287425\) | \(-928973481143317960442400000000\) | \([2]\) | \(339738624\) | \(4.4450\) | |
| 323400.ij4 | 323400ij2 | \([0, 1, 0, -168584908, 448335982688]\) | \(1125982298608534096/467044181552025\) | \(219789123661656756900000000\) | \([2, 2]\) | \(84934656\) | \(3.7518\) | |
| 323400.ij5 | 323400ij1 | \([0, 1, 0, -78908783, -264947915562]\) | \(1847444944806639616/38285567941005\) | \(1126064695672824311250000\) | \([2]\) | \(42467328\) | \(3.4052\) | \(\Gamma_0(N)\)-optimal |
| 323400.ij6 | 323400ij4 | \([0, 1, 0, 557717592, 3283820942688]\) | \(10191978981888338876/8372623608979245\) | \(-15760492719564787120080000000\) | \([2]\) | \(169869312\) | \(4.0984\) |
Rank
sage: E.rank()
The elliptic curves in class 323400.ij have rank \(0\).
Complex multiplication
The elliptic curves in class 323400.ij do not have complex multiplication.Modular form 323400.2.a.ij
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
