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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 122010h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122010.i2 | 122010h1 | \([1, 1, 0, -1555677, -733336659]\) | \(3539111138359094089/76880750864400\) | \(9044943458445795600\) | \([2]\) | \(3317760\) | \(2.4258\) | \(\Gamma_0(N)\)-optimal |
122010.i4 | 122010h2 | \([1, 1, 0, 125023, -2232857199]\) | \(1836960390293111/18308926676226060\) | \(-2154026914531319732940\) | \([2]\) | \(6635520\) | \(2.7724\) | |
122010.i1 | 122010h3 | \([1, 1, 0, -14845212, 21707761104]\) | \(3075351278706579064249/48414348864000000\) | \(5695899729500736000000\) | \([2]\) | \(9953280\) | \(2.9751\) | |
122010.i3 | 122010h4 | \([1, 1, 0, -1125212, 60291145104]\) | \(-1339180572778744249/13347013802416056000\) | \(-1570262826840446572344000\) | \([2]\) | \(19906560\) | \(3.3217\) |
Rank
sage: E.rank()
The elliptic curves in class 122010h have rank \(2\).
Complex multiplication
The elliptic curves in class 122010h do not have complex multiplication.Modular form 122010.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.