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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 139650.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.l1 | 139650iq3 | \([1, 1, 0, -107251225, 427470461125]\) | \(74220219816682217473/16416\) | \(30176968500000\) | \([2]\) | \(11796480\) | \(2.8791\) | |
139650.l2 | 139650iq2 | \([1, 1, 0, -6703225, 6677081125]\) | \(18120364883707393/269485056\) | \(495385114896000000\) | \([2, 2]\) | \(5898240\) | \(2.5325\) | |
139650.l3 | 139650iq4 | \([1, 1, 0, -6507225, 7086133125]\) | \(-16576888679672833/2216253521952\) | \(-4074062665689544500000\) | \([2]\) | \(11796480\) | \(2.8791\) | |
139650.l4 | 139650iq1 | \([1, 1, 0, -431225, 97753125]\) | \(4824238966273/537919488\) | \(988838903808000000\) | \([2]\) | \(2949120\) | \(2.1860\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139650.l have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.l do not have complex multiplication.Modular form 139650.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.