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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 47040.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.r1 | 47040e4 | \([0, -1, 0, -732321, 241456545]\) | \(5633270409316/14175\) | \(109292720947200\) | \([2]\) | \(393216\) | \(1.9320\) | |
| 47040.r2 | 47040e3 | \([0, -1, 0, -128641, -12956159]\) | \(30534944836/8203125\) | \(63248102400000000\) | \([2]\) | \(393216\) | \(1.9320\) | |
| 47040.r3 | 47040e2 | \([0, -1, 0, -46321, 3688945]\) | \(5702413264/275625\) | \(531284060160000\) | \([2, 2]\) | \(196608\) | \(1.5854\) | |
| 47040.r4 | 47040e1 | \([0, -1, 0, 1699, 221901]\) | \(4499456/180075\) | \(-21694099123200\) | \([2]\) | \(98304\) | \(1.2389\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.r have rank \(2\).
Complex multiplication
The elliptic curves in class 47040.r do not have complex multiplication.Modular form 47040.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
