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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 44688m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44688.bm4 | 44688m1 | \([0, -1, 0, -213607, -14812622]\) | \(572616640141312/280535480757\) | \(528075500409284688\) | \([2]\) | \(589824\) | \(2.0940\) | \(\Gamma_0(N)\)-optimal |
44688.bm2 | 44688m2 | \([0, -1, 0, -1821052, 936151840]\) | \(22174957026242512/278654127129\) | \(8392545127065528576\) | \([2, 2]\) | \(1179648\) | \(2.4405\) | |
44688.bm3 | 44688m3 | \([0, -1, 0, -312832, 2438338960]\) | \(-28104147578308/21301741002339\) | \(-2566275611836601355264\) | \([2]\) | \(2359296\) | \(2.7871\) | |
44688.bm1 | 44688m4 | \([0, -1, 0, -29048392, 60269971168]\) | \(22501000029889239268/3620708343\) | \(436196061025901568\) | \([4]\) | \(2359296\) | \(2.7871\) |
Rank
sage: E.rank()
The elliptic curves in class 44688m have rank \(1\).
Complex multiplication
The elliptic curves in class 44688m do not have complex multiplication.Modular form 44688.2.a.m
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.