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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 7406d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7406.a5 | 7406d1 | \([1, 0, 1, -276, -3586]\) | \(-15625/28\) | \(-4145004892\) | \([2]\) | \(4224\) | \(0.53566\) | \(\Gamma_0(N)\)-optimal |
| 7406.a4 | 7406d2 | \([1, 0, 1, -5566, -160170]\) | \(128787625/98\) | \(14507517122\) | \([2]\) | \(8448\) | \(0.88224\) | |
| 7406.a6 | 7406d3 | \([1, 0, 1, 2369, 74706]\) | \(9938375/21952\) | \(-3249683835328\) | \([2]\) | \(12672\) | \(1.0850\) | |
| 7406.a3 | 7406d4 | \([1, 0, 1, -18791, 811074]\) | \(4956477625/941192\) | \(139330194439688\) | \([2]\) | \(25344\) | \(1.4315\) | |
| 7406.a2 | 7406d5 | \([1, 0, 1, -90206, 10450512]\) | \(-548347731625/1835008\) | \(-271647040602112\) | \([2]\) | \(38016\) | \(1.6343\) | |
| 7406.a1 | 7406d6 | \([1, 0, 1, -1444446, 668069456]\) | \(2251439055699625/25088\) | \(3713924383232\) | \([2]\) | \(76032\) | \(1.9808\) |
Rank
sage: E.rank()
The elliptic curves in class 7406d have rank \(2\).
Complex multiplication
The elliptic curves in class 7406d do not have complex multiplication.Modular form 7406.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
