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SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 705600.hr
sage: E.isogeny_class().curves
| LMFDB label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height |
|---|---|---|---|---|---|---|
| 705600.hr1 | \([0, 0, 0, -3704414700, 86781571814000]\) | \(128025588102048008/7875\) | \(345808999872000000000\) | \([2]\) | \(226492416\) | \(3.8495\) |
| 705600.hr2 | \([0, 0, 0, -259322700, 1010005346000]\) | \(43919722445768/15380859375\) | \(675408202875000000000000000\) | \([2]\) | \(226492416\) | \(3.8495\) |
| 705600.hr3 | \([0, 0, 0, -231539700, 1355792564000]\) | \(250094631024064/62015625\) | \(340405734249000000000000\) | \([2, 2]\) | \(113246208\) | \(3.5029\) |
| 705600.hr4 | \([0, 0, 0, -12748575, 26417688500]\) | \(-2671731885376/1969120125\) | \(-168883794904285125000000\) | \([2]\) | \(56623104\) | \(3.1564\) |
Rank
sage: E.rank()
The elliptic curves in class 705600.hr have rank \(1\).
Complex multiplication
The elliptic curves in class 705600.hr do not have complex multiplication.Modular form 705600.2.a.hr
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
