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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 16245.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16245.c1 | 16245d7 | \([1, -1, 1, -7017908, -7154068534]\) | \(1114544804970241/405\) | \(13890061135845\) | \([2]\) | \(221184\) | \(2.3124\) | |
16245.c2 | 16245d5 | \([1, -1, 1, -438683, -111666094]\) | \(272223782641/164025\) | \(5625474760017225\) | \([2, 2]\) | \(110592\) | \(1.9658\) | |
16245.c3 | 16245d8 | \([1, -1, 1, -357458, -154357954]\) | \(-147281603041/215233605\) | \(-7381747980094602645\) | \([2]\) | \(221184\) | \(2.3124\) | |
16245.c4 | 16245d4 | \([1, -1, 1, -259988, 51089312]\) | \(56667352321/15\) | \(514446708735\) | \([2]\) | \(55296\) | \(1.6192\) | |
16245.c5 | 16245d3 | \([1, -1, 1, -32558, -1037644]\) | \(111284641/50625\) | \(1736257641980625\) | \([2, 2]\) | \(55296\) | \(1.6192\) | |
16245.c6 | 16245d2 | \([1, -1, 1, -16313, 794792]\) | \(13997521/225\) | \(7716700631025\) | \([2, 2]\) | \(27648\) | \(1.2727\) | |
16245.c7 | 16245d1 | \([1, -1, 1, -68, 34526]\) | \(-1/15\) | \(-514446708735\) | \([2]\) | \(13824\) | \(0.92610\) | \(\Gamma_0(N)\)-optimal |
16245.c8 | 16245d6 | \([1, -1, 1, 113647, -7880038]\) | \(4733169839/3515625\) | \(-120573447359765625\) | \([2]\) | \(110592\) | \(1.9658\) |
Rank
sage: E.rank()
The elliptic curves in class 16245.c have rank \(1\).
Complex multiplication
The elliptic curves in class 16245.c do not have complex multiplication.Modular form 16245.2.a.c
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.