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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 53040b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53040.j3 | 53040b1 | \([0, -1, 0, -476, -3264]\) | \(46689225424/7249905\) | \(1855975680\) | \([2]\) | \(24576\) | \(0.50172\) | \(\Gamma_0(N)\)-optimal |
53040.j2 | 53040b2 | \([0, -1, 0, -2096, 34320]\) | \(994958062276/98903025\) | \(101276697600\) | \([2, 2]\) | \(49152\) | \(0.84829\) | |
53040.j4 | 53040b3 | \([0, -1, 0, 2584, 161616]\) | \(931329171502/6107473125\) | \(-12508104960000\) | \([2]\) | \(98304\) | \(1.1949\) | |
53040.j1 | 53040b4 | \([0, -1, 0, -32696, 2286480]\) | \(1887517194957938/21849165\) | \(44747089920\) | \([2]\) | \(98304\) | \(1.1949\) |
Rank
sage: E.rank()
The elliptic curves in class 53040b have rank \(2\).
Complex multiplication
The elliptic curves in class 53040b do not have complex multiplication.Modular form 53040.2.a.b
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.