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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 40432r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40432.b5 | 40432r1 | \([0, 1, 0, -3008, 127540]\) | \(-15625/28\) | \(-5395598000128\) | \([2]\) | \(57024\) | \(1.1333\) | \(\Gamma_0(N)\)-optimal |
| 40432.b4 | 40432r2 | \([0, 1, 0, -60768, 5741812]\) | \(128787625/98\) | \(18884593000448\) | \([2]\) | \(114048\) | \(1.4799\) | |
| 40432.b6 | 40432r3 | \([0, 1, 0, 25872, -2679596]\) | \(9938375/21952\) | \(-4230148832100352\) | \([2]\) | \(171072\) | \(1.6826\) | |
| 40432.b3 | 40432r4 | \([0, 1, 0, -205168, -29387820]\) | \(4956477625/941192\) | \(181367631176302592\) | \([2]\) | \(342144\) | \(2.0292\) | |
| 40432.b2 | 40432r5 | \([0, 1, 0, -984928, -377645964]\) | \(-548347731625/1835008\) | \(-353605910536388608\) | \([2]\) | \(513216\) | \(2.2319\) | |
| 40432.b1 | 40432r6 | \([0, 1, 0, -15771488, -24113032076]\) | \(2251439055699625/25088\) | \(4834455808114688\) | \([2]\) | \(1026432\) | \(2.5785\) |
Rank
sage: E.rank()
The elliptic curves in class 40432r have rank \(1\).
Complex multiplication
The elliptic curves in class 40432r do not have complex multiplication.Modular form 40432.2.a.r
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.
