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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1170.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1170.k1 | 1170k4 | \([1, -1, 1, -7853198, 8472578717]\) | \(73474353581350183614361/576510977802240\) | \(420276502817832960\) | \([6]\) | \(34560\) | \(2.5547\) | |
1170.k2 | 1170k3 | \([1, -1, 1, -480398, 138365597]\) | \(-16818951115904497561/1592332281446400\) | \(-1160810233174425600\) | \([6]\) | \(17280\) | \(2.2081\) | |
1170.k3 | 1170k2 | \([1, -1, 1, -144023, -755953]\) | \(453198971846635561/261896250564000\) | \(190922366661156000\) | \([2]\) | \(11520\) | \(2.0054\) | |
1170.k4 | 1170k1 | \([1, -1, 1, 35977, -107953]\) | \(7064514799444439/4094064000000\) | \(-2984572656000000\) | \([2]\) | \(5760\) | \(1.6588\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1170.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1170.k do not have complex multiplication.Modular form 1170.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.