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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 720h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
720.c7 | 720h1 | \([0, 0, 0, -3, 322]\) | \(-1/15\) | \(-44789760\) | \([2]\) | \(128\) | \(0.14703\) | \(\Gamma_0(N)\)-optimal |
720.c6 | 720h2 | \([0, 0, 0, -723, 7378]\) | \(13997521/225\) | \(671846400\) | \([2, 2]\) | \(256\) | \(0.49360\) | |
720.c5 | 720h3 | \([0, 0, 0, -1443, -9758]\) | \(111284641/50625\) | \(151165440000\) | \([2, 2]\) | \(512\) | \(0.84018\) | |
720.c4 | 720h4 | \([0, 0, 0, -11523, 476098]\) | \(56667352321/15\) | \(44789760\) | \([2]\) | \(512\) | \(0.84018\) | |
720.c2 | 720h5 | \([0, 0, 0, -19443, -1042958]\) | \(272223782641/164025\) | \(489776025600\) | \([2, 2]\) | \(1024\) | \(1.1867\) | |
720.c8 | 720h6 | \([0, 0, 0, 5037, -73262]\) | \(4733169839/3515625\) | \(-10497600000000\) | \([2]\) | \(1024\) | \(1.1867\) | |
720.c1 | 720h7 | \([0, 0, 0, -311043, -66769598]\) | \(1114544804970241/405\) | \(1209323520\) | \([2]\) | \(2048\) | \(1.5333\) | |
720.c3 | 720h8 | \([0, 0, 0, -15843, -1441118]\) | \(-147281603041/215233605\) | \(-642684100792320\) | \([2]\) | \(2048\) | \(1.5333\) |
Rank
sage: E.rank()
The elliptic curves in class 720h have rank \(1\).
Complex multiplication
The elliptic curves in class 720h do not have complex multiplication.Modular form 720.2.a.h
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.