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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 30752f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
30752.c3 | 30752f1 | \([0, 0, 0, -961, 0]\) | \(1728\) | \(56800235584\) | \([2, 2]\) | \(15360\) | \(0.75303\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
30752.c4 | 30752f2 | \([0, 0, 0, 3844, 0]\) | \(1728\) | \(-3635215077376\) | \([2]\) | \(30720\) | \(1.0996\) | \(-4\) | |
30752.c1 | 30752f3 | \([0, 0, 0, -10571, -417074]\) | \(287496\) | \(454401884672\) | \([2]\) | \(30720\) | \(1.0996\) | \(-16\) | |
30752.c2 | 30752f4 | \([0, 0, 0, -10571, 417074]\) | \(287496\) | \(454401884672\) | \([2]\) | \(30720\) | \(1.0996\) | \(-16\) |
Rank
sage: E.rank()
The elliptic curves in class 30752f have rank \(1\).
Complex multiplication
Each elliptic curve in class 30752f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 30752.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.