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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 4410m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4410.b4 | 4410m1 | \([1, -1, 0, 1020, -20224]\) | \(1367631/2800\) | \(-240145138800\) | \([2]\) | \(6144\) | \(0.86810\) | \(\Gamma_0(N)\)-optimal |
4410.b3 | 4410m2 | \([1, -1, 0, -7800, -212500]\) | \(611960049/122500\) | \(10506349822500\) | \([2, 2]\) | \(12288\) | \(1.2147\) | |
4410.b1 | 4410m3 | \([1, -1, 0, -118050, -15581350]\) | \(2121328796049/120050\) | \(10296222826050\) | \([2]\) | \(24576\) | \(1.5612\) | |
4410.b2 | 4410m4 | \([1, -1, 0, -38670, 2744846]\) | \(74565301329/5468750\) | \(469033474218750\) | \([2]\) | \(24576\) | \(1.5612\) |
Rank
sage: E.rank()
The elliptic curves in class 4410m have rank \(1\).
Complex multiplication
The elliptic curves in class 4410m do not have complex multiplication.Modular form 4410.2.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.