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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 407330h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
407330.h4 | 407330h1 | \([1, -1, 0, -6395180, 7493760976]\) | \(-195395722614328041/50730248800000\) | \(-7509897480299183200000\) | \([2]\) | \(31539200\) | \(2.9146\) | \(\Gamma_0(N)\)-optimal* |
407330.h3 | 407330h2 | \([1, -1, 0, -108005500, 432042000000]\) | \(941226862950447171561/45393906250000\) | \(6719927266901406250000\) | \([2, 2]\) | \(63078400\) | \(3.2612\) | \(\Gamma_0(N)\)-optimal* |
407330.h1 | 407330h3 | \([1, -1, 0, -1728068000, 27650064037500]\) | \(3855131356812007128171561/8967612500\) | \(1327528488645012500\) | \([2]\) | \(126156800\) | \(3.6078\) | \(\Gamma_0(N)\)-optimal* |
407330.h2 | 407330h4 | \([1, -1, 0, -113708120, 383887936196]\) | \(1098325674097093229481/205612182617187500\) | \(30437982242965698242187500\) | \([2]\) | \(126156800\) | \(3.6078\) |
Rank
sage: E.rank()
The elliptic curves in class 407330h have rank \(0\).
Complex multiplication
The elliptic curves in class 407330h do not have complex multiplication.Modular form 407330.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.