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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 27378.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27378.p1 | 27378s3 | \([1, -1, 1, -182045, 29941669]\) | \(-189613868625/128\) | \(-450399201408\) | \([]\) | \(96768\) | \(1.5518\) | |
27378.p2 | 27378s4 | \([1, -1, 1, -144020, 42769783]\) | \(-1159088625/2097152\) | \(-597726581785362432\) | \([]\) | \(290304\) | \(2.1012\) | |
27378.p3 | 27378s2 | \([1, -1, 1, -7130, -241055]\) | \(-140625/8\) | \(-2280145957128\) | \([]\) | \(41472\) | \(1.1282\) | |
27378.p4 | 27378s1 | \([1, -1, 1, 475, -737]\) | \(3375/2\) | \(-7037487522\) | \([]\) | \(13824\) | \(0.57889\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27378.p have rank \(1\).
Complex multiplication
The elliptic curves in class 27378.p do not have complex multiplication.Modular form 27378.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.