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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 7350w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7350.bd7 | 7350w1 | \([1, 0, 1, -50251, 1521398]\) | \(7633736209/3870720\) | \(7115411520000000\) | \([2]\) | \(55296\) | \(1.7342\) | \(\Gamma_0(N)\)-optimal |
| 7350.bd5 | 7350w2 | \([1, 0, 1, -442251, -112158602]\) | \(5203798902289/57153600\) | \(105063498225000000\) | \([2, 2]\) | \(110592\) | \(2.0808\) | |
| 7350.bd4 | 7350w3 | \([1, 0, 1, -3284251, 2290605398]\) | \(2131200347946769/2058000\) | \(3783150656250000\) | \([2]\) | \(165888\) | \(2.2836\) | |
| 7350.bd2 | 7350w4 | \([1, 0, 1, -7057251, -7216668602]\) | \(21145699168383889/2593080\) | \(4766769826875000\) | \([2]\) | \(221184\) | \(2.4274\) | |
| 7350.bd6 | 7350w5 | \([1, 0, 1, -99251, -281600602]\) | \(-58818484369/18600435000\) | \(-34192540270546875000\) | \([2]\) | \(221184\) | \(2.4274\) | |
| 7350.bd3 | 7350w6 | \([1, 0, 1, -3308751, 2254688398]\) | \(2179252305146449/66177562500\) | \(121651938290039062500\) | \([2, 2]\) | \(331776\) | \(2.6301\) | |
| 7350.bd1 | 7350w7 | \([1, 0, 1, -7902501, -5380124102]\) | \(29689921233686449/10380965400750\) | \(19082971850513074218750\) | \([2]\) | \(663552\) | \(2.9767\) | |
| 7350.bd8 | 7350w8 | \([1, 0, 1, 892999, 7590910898]\) | \(42841933504271/13565917968750\) | \(-24937760673522949218750\) | \([2]\) | \(663552\) | \(2.9767\) |
Rank
sage: E.rank()
The elliptic curves in class 7350w have rank \(1\).
Complex multiplication
The elliptic curves in class 7350w do not have complex multiplication.Modular form 7350.2.a.w
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.
