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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1155.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1155.e1 | 1155e5 | \([1, 1, 1, -13250, -592540]\) | \(257260669489908001/14267882475\) | \(14267882475\) | \([2]\) | \(2048\) | \(1.0139\) | |
1155.e2 | 1155e3 | \([1, 1, 1, -875, -8440]\) | \(74093292126001/14707625625\) | \(14707625625\) | \([2, 2]\) | \(1024\) | \(0.66729\) | |
1155.e3 | 1155e2 | \([1, 1, 1, -270, 1482]\) | \(2177286259681/161417025\) | \(161417025\) | \([2, 4]\) | \(512\) | \(0.32072\) | |
1155.e4 | 1155e1 | \([1, 1, 1, -265, 1550]\) | \(2058561081361/12705\) | \(12705\) | \([4]\) | \(256\) | \(-0.025854\) | \(\Gamma_0(N)\)-optimal |
1155.e5 | 1155e4 | \([1, 1, 1, 255, 7152]\) | \(1833318007919/22507682505\) | \(-22507682505\) | \([4]\) | \(1024\) | \(0.66729\) | |
1155.e6 | 1155e6 | \([1, 1, 1, 1820, -47248]\) | \(666688497209279/1381398046875\) | \(-1381398046875\) | \([2]\) | \(2048\) | \(1.0139\) |
Rank
sage: E.rank()
The elliptic curves in class 1155.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1155.e do not have complex multiplication.Modular form 1155.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.