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SageMath
E = EllipticCurve("fh1")
E.isogeny_class()
Elliptic curves in class 221760fh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.r4 | 221760fh1 | \([0, 0, 0, -2568, -1340008]\) | \(-2508888064/1037680875\) | \(-774624622464000\) | \([2]\) | \(983040\) | \(1.5361\) | \(\Gamma_0(N)\)-optimal |
221760.r3 | 221760fh2 | \([0, 0, 0, -198588, -33722512]\) | \(72516235474384/833765625\) | \(9958443264000000\) | \([2, 2]\) | \(1966080\) | \(1.8827\) | |
221760.r2 | 221760fh3 | \([0, 0, 0, -364908, 31009232]\) | \(112477694831716/56396484375\) | \(2694384000000000000\) | \([2]\) | \(3932160\) | \(2.2292\) | |
221760.r1 | 221760fh4 | \([0, 0, 0, -3168588, -2170934512]\) | \(73639964854838596/9904125\) | \(473176940544000\) | \([2]\) | \(3932160\) | \(2.2292\) |
Rank
sage: E.rank()
The elliptic curves in class 221760fh have rank \(1\).
Complex multiplication
The elliptic curves in class 221760fh do not have complex multiplication.Modular form 221760.2.a.fh
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.