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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 75712cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.m5 | 75712cy1 | \([0, 1, 0, -5633, -331265]\) | \(-15625/28\) | \(-35428932517888\) | \([2]\) | \(138240\) | \(1.2901\) | \(\Gamma_0(N)\)-optimal |
| 75712.m4 | 75712cy2 | \([0, 1, 0, -113793, -14803073]\) | \(128787625/98\) | \(124001263812608\) | \([2]\) | \(276480\) | \(1.6367\) | |
| 75712.m6 | 75712cy3 | \([0, 1, 0, 48447, 6904639]\) | \(9938375/21952\) | \(-27776283094024192\) | \([2]\) | \(414720\) | \(1.8394\) | |
| 75712.m3 | 75712cy4 | \([0, 1, 0, -384193, 75002175]\) | \(4956477625/941192\) | \(1190908137656287232\) | \([2]\) | \(829440\) | \(2.1860\) | |
| 75712.m2 | 75712cy5 | \([0, 1, 0, -1844353, 966251391]\) | \(-548347731625/1835008\) | \(-2321870521492307968\) | \([2]\) | \(1244160\) | \(2.3887\) | |
| 75712.m1 | 75712cy6 | \([0, 1, 0, -29533313, 61765669759]\) | \(2251439055699625/25088\) | \(31744323536027648\) | \([2]\) | \(2488320\) | \(2.7353\) |
Rank
sage: E.rank()
The elliptic curves in class 75712cy have rank \(1\).
Complex multiplication
The elliptic curves in class 75712cy do not have complex multiplication.Modular form 75712.2.a.cy
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.
