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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 444675ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 444675.ca4 | 444675ca1 | \([1, 1, 1, -39282713, 94748641406]\) | \(2058561081361/12705\) | \(41375099047355390625\) | \([2]\) | \(35389440\) | \(2.9508\) | \(\Gamma_0(N)\)-optimal |
| 444675.ca3 | 444675ca2 | \([1, 1, 1, -40023838, 90986690906]\) | \(2177286259681/161417025\) | \(525670633396650237890625\) | \([2, 2]\) | \(70778880\) | \(3.2973\) | |
| 444675.ca5 | 444675ca3 | \([1, 1, 1, 37794287, 402103554656]\) | \(1833318007919/22507682505\) | \(-73298511843431963171015625\) | \([2]\) | \(141557760\) | \(3.6439\) | |
| 444675.ca2 | 444675ca4 | \([1, 1, 1, -129699963, -460880182344]\) | \(74093292126001/14707625625\) | \(47896849034694784072265625\) | \([2, 2]\) | \(141557760\) | \(3.6439\) | |
| 444675.ca6 | 444675ca5 | \([1, 1, 1, 269766412, -2739436385344]\) | \(666688497209279/1381398046875\) | \(-4498660449687242010498046875\) | \([2]\) | \(283115520\) | \(3.9905\) | |
| 444675.ca1 | 444675ca6 | \([1, 1, 1, -1963984338, -33500010344844]\) | \(257260669489908001/14267882475\) | \(46464781629213000629296875\) | \([2]\) | \(283115520\) | \(3.9905\) |
Rank
sage: E.rank()
The elliptic curves in class 444675ca have rank \(1\).
Complex multiplication
The elliptic curves in class 444675ca do not have complex multiplication.Modular form 444675.2.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.
