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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 61347z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61347.i5 | 61347z1 | \([1, 0, 0, -491202, -190117773]\) | \(-1532808577/938223\) | \(-8022732280967445327\) | \([4]\) | \(1290240\) | \(2.3294\) | \(\Gamma_0(N)\)-optimal |
| 61347.i4 | 61347z2 | \([1, 0, 0, -8773047, -10000791360]\) | \(8732907467857/1656369\) | \(14163589088621539281\) | \([2, 2]\) | \(2580480\) | \(2.6759\) | |
| 61347.i3 | 61347z3 | \([1, 0, 0, -9693252, -7774815465]\) | \(11779205551777/3763454409\) | \(32181248141469095195241\) | \([2, 2]\) | \(5160960\) | \(3.0225\) | |
| 61347.i1 | 61347z4 | \([1, 0, 0, -140362362, -640076749443]\) | \(35765103905346817/1287\) | \(11005119726978663\) | \([2]\) | \(5160960\) | \(3.0225\) | |
| 61347.i6 | 61347z5 | \([1, 0, 0, 27421683, -52973383308]\) | \(266679605718863/296110251723\) | \(-2532034788342971977606827\) | \([2]\) | \(10321920\) | \(3.3691\) | |
| 61347.i2 | 61347z6 | \([1, 0, 0, -61531467, 179889890478]\) | \(3013001140430737/108679952667\) | \(929320816645461583340283\) | \([2]\) | \(10321920\) | \(3.3691\) |
Rank
sage: E.rank()
The elliptic curves in class 61347z have rank \(1\).
Complex multiplication
The elliptic curves in class 61347z do not have complex multiplication.Modular form 61347.2.a.z
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
