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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 4410.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4410.l1 | 4410t5 | \([1, -1, 0, -7408809, -7760095785]\) | \(524388516989299201/3150\) | \(270163281150\) | \([2]\) | \(98304\) | \(2.2590\) | |
4410.l2 | 4410t3 | \([1, -1, 0, -463059, -121159935]\) | \(128031684631201/9922500\) | \(851014335622500\) | \([2, 2]\) | \(49152\) | \(1.9124\) | |
4410.l3 | 4410t6 | \([1, -1, 0, -432189, -138033477]\) | \(-104094944089921/35880468750\) | \(-3077328624349218750\) | \([2]\) | \(98304\) | \(2.2590\) | |
4410.l4 | 4410t4 | \([1, -1, 0, -163179, 24020793]\) | \(5602762882081/345888060\) | \(29665477206415260\) | \([2]\) | \(49152\) | \(1.9124\) | |
4410.l5 | 4410t2 | \([1, -1, 0, -30879, -1618947]\) | \(37966934881/8643600\) | \(741328043475600\) | \([2, 2]\) | \(24576\) | \(1.5659\) | |
4410.l6 | 4410t1 | \([1, -1, 0, 4401, -158355]\) | \(109902239/188160\) | \(-16137753327360\) | \([2]\) | \(12288\) | \(1.2193\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4410.l have rank \(0\).
Complex multiplication
The elliptic curves in class 4410.l do not have complex multiplication.Modular form 4410.2.a.l
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.