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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 73920ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73920.eq4 | 73920ca1 | \([0, 1, 0, -16961, 844575]\) | \(2058561081361/12705\) | \(3330539520\) | \([2]\) | \(131072\) | \(1.0139\) | \(\Gamma_0(N)\)-optimal |
73920.eq3 | 73920ca2 | \([0, 1, 0, -17281, 810719]\) | \(2177286259681/161417025\) | \(42314504601600\) | \([2, 2]\) | \(262144\) | \(1.3604\) | |
73920.eq5 | 73920ca3 | \([0, 1, 0, 16319, 3612959]\) | \(1833318007919/22507682505\) | \(-5900253922590720\) | \([2]\) | \(524288\) | \(1.7070\) | |
73920.eq2 | 73920ca4 | \([0, 1, 0, -56001, -4153185]\) | \(74093292126001/14707625625\) | \(3855515811840000\) | \([2, 2]\) | \(524288\) | \(1.7070\) | |
73920.eq6 | 73920ca5 | \([0, 1, 0, 116479, -24540321]\) | \(666688497209279/1381398046875\) | \(-362125209600000000\) | \([2]\) | \(1048576\) | \(2.0536\) | |
73920.eq1 | 73920ca6 | \([0, 1, 0, -848001, -300836385]\) | \(257260669489908001/14267882475\) | \(3740239783526400\) | \([2]\) | \(1048576\) | \(2.0536\) |
Rank
sage: E.rank()
The elliptic curves in class 73920ca have rank \(0\).
Complex multiplication
The elliptic curves in class 73920ca do not have complex multiplication.Modular form 73920.2.a.ca
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.