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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1176e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1176.f4 | 1176e1 | \([0, 1, 0, -359, -17130]\) | \(-2725888/64827\) | \(-122029307568\) | \([2]\) | \(1152\) | \(0.80800\) | \(\Gamma_0(N)\)-optimal |
1176.f3 | 1176e2 | \([0, 1, 0, -12364, -530944]\) | \(6940769488/35721\) | \(1075850221824\) | \([2, 2]\) | \(2304\) | \(1.1546\) | |
1176.f1 | 1176e3 | \([0, 1, 0, -197584, -33870544]\) | \(7080974546692/189\) | \(22769316864\) | \([2]\) | \(4608\) | \(1.5011\) | |
1176.f2 | 1176e4 | \([0, 1, 0, -19224, 116640]\) | \(6522128932/3720087\) | \(448168463834112\) | \([4]\) | \(4608\) | \(1.5011\) |
Rank
sage: E.rank()
The elliptic curves in class 1176e have rank \(1\).
Complex multiplication
The elliptic curves in class 1176e do not have complex multiplication.Modular form 1176.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.