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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 76230.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76230.ek1 | 76230en4 | \([1, -1, 1, -28440347, -56716844929]\) | \(1969902499564819009/63690429687500\) | \(82254149873252929687500\) | \([2]\) | \(9953280\) | \(3.1709\) | |
| 76230.ek2 | 76230en2 | \([1, -1, 1, -3894287, 2932642199]\) | \(5057359576472449/51765560000\) | \(66853562637347640000\) | \([2]\) | \(3317760\) | \(2.6216\) | |
| 76230.ek3 | 76230en1 | \([1, -1, 1, -61007, 112881431]\) | \(-19443408769/4249907200\) | \(-5488619020022476800\) | \([2]\) | \(1658880\) | \(2.2750\) | \(\Gamma_0(N)\)-optimal |
| 76230.ek4 | 76230en3 | \([1, -1, 1, 548833, -3040479241]\) | \(14156681599871/3100231750000\) | \(-4003850001601815750000\) | \([2]\) | \(4976640\) | \(2.8243\) |
Rank
sage: E.rank()
The elliptic curves in class 76230.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 76230.ek do not have complex multiplication.Modular form 76230.2.a.ek
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
