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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 333270i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.i3 | 333270i1 | \([1, -1, 0, -1417290, -325906444]\) | \(2917464019569/1262240000\) | \(136218622167361440000\) | \([2]\) | \(12976128\) | \(2.5596\) | \(\Gamma_0(N)\)-optimal |
333270.i2 | 333270i2 | \([1, -1, 0, -10939290, 13703808356]\) | \(1341518286067569/24894528400\) | \(2686571775695786000400\) | \([2, 2]\) | \(25952256\) | \(2.9062\) | |
333270.i1 | 333270i3 | \([1, -1, 0, -174241590, 885313504376]\) | \(5421065386069310769/1919709260\) | \(207171496988785830060\) | \([2]\) | \(51904512\) | \(3.2528\) | |
333270.i4 | 333270i4 | \([1, -1, 0, 11010, 39881595536]\) | \(1367631/6366992112460\) | \(-687114093127898972089260\) | \([2]\) | \(51904512\) | \(3.2528\) |
Rank
sage: E.rank()
The elliptic curves in class 333270i have rank \(1\).
Complex multiplication
The elliptic curves in class 333270i do not have complex multiplication.Modular form 333270.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.