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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 37570k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37570.l3 | 37570k1 | \([1, -1, 1, -457108, 117956727]\) | \(437608510454961/4707123200\) | \(113618511031500800\) | \([4]\) | \(589824\) | \(2.0882\) | \(\Gamma_0(N)\)-optimal |
37570.l2 | 37570k2 | \([1, -1, 1, -827028, -99852169]\) | \(2591733435976881/1320660640000\) | \(31877537323584160000\) | \([2, 2]\) | \(1179648\) | \(2.4347\) | |
37570.l4 | 37570k3 | \([1, -1, 1, 3080252, -775030153]\) | \(133902615693854799/88219056250000\) | \(-2129393557349256250000\) | \([2]\) | \(2359296\) | \(2.7813\) | |
37570.l1 | 37570k4 | \([1, -1, 1, -10653028, -13368882569]\) | \(5539229398623592881/5546968902800\) | \(133890344632189293200\) | \([2]\) | \(2359296\) | \(2.7813\) |
Rank
sage: E.rank()
The elliptic curves in class 37570k have rank \(1\).
Complex multiplication
The elliptic curves in class 37570k do not have complex multiplication.Modular form 37570.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.