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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 80850l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.e5 | 80850l1 | \([1, 1, 0, 42850, -39967500]\) | \(4733169839/378470400\) | \(-695729126400000000\) | \([2]\) | \(1179648\) | \(2.1032\) | \(\Gamma_0(N)\)-optimal |
80850.e4 | 80850l2 | \([1, 1, 0, -1525150, -700095500]\) | \(213429068128081/8537760000\) | \(15694670722500000000\) | \([2, 2]\) | \(2359296\) | \(2.4497\) | |
80850.e3 | 80850l3 | \([1, 1, 0, -3975150, 2110054500]\) | \(3778993806976081/1138958528400\) | \(2093708311058306250000\) | \([2, 2]\) | \(4718592\) | \(2.7963\) | |
80850.e2 | 80850l4 | \([1, 1, 0, -24163150, -45727077500]\) | \(848742840525560401/1443750000\) | \(2653995996093750000\) | \([2]\) | \(4718592\) | \(2.7963\) | |
80850.e6 | 80850l5 | \([1, 1, 0, 10847350, 14160747000]\) | \(76786760064334319/91531319653620\) | \(-168258878530136552812500\) | \([2]\) | \(9437184\) | \(3.1429\) | |
80850.e1 | 80850l6 | \([1, 1, 0, -57997650, 169957962000]\) | \(11736717412386894481/1890645330420\) | \(3475508319977852812500\) | \([2]\) | \(9437184\) | \(3.1429\) |
Rank
sage: E.rank()
The elliptic curves in class 80850l have rank \(0\).
Complex multiplication
The elliptic curves in class 80850l do not have complex multiplication.Modular form 80850.2.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 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.