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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 121275dh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.bh4 | 121275dh1 | \([1, -1, 1, -2921855, 1923086022]\) | \(2058561081361/12705\) | \(17025915114140625\) | \([4]\) | \(2359296\) | \(2.3011\) | \(\Gamma_0(N)\)-optimal |
| 121275.bh3 | 121275dh2 | \([1, -1, 1, -2976980, 1846793022]\) | \(2177286259681/161417025\) | \(216314251525156640625\) | \([2, 2]\) | \(4718592\) | \(2.6477\) | |
| 121275.bh5 | 121275dh3 | \([1, -1, 1, 2811145, 8155849272]\) | \(1833318007919/22507682505\) | \(-30162447205522079765625\) | \([2]\) | \(9437184\) | \(2.9943\) | |
| 121275.bh2 | 121275dh4 | \([1, -1, 1, -9647105, -9345676728]\) | \(74093292126001/14707625625\) | \(19709624984007041015625\) | \([2, 2]\) | \(9437184\) | \(2.9943\) | |
| 121275.bh6 | 121275dh5 | \([1, -1, 1, 20065270, -55578132228]\) | \(666688497209279/1381398046875\) | \(-1851205500585076904296875\) | \([2]\) | \(18874368\) | \(3.3408\) | |
| 121275.bh1 | 121275dh6 | \([1, -1, 1, -146081480, -679511326728]\) | \(257260669489908001/14267882475\) | \(19120327105697335546875\) | \([2]\) | \(18874368\) | \(3.3408\) |
Rank
sage: E.rank()
The elliptic curves in class 121275dh have rank \(1\).
Complex multiplication
The elliptic curves in class 121275dh do not have complex multiplication.Modular form 121275.2.a.dh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
