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SageMath
E = EllipticCurve("it1")
E.isogeny_class()
Elliptic curves in class 428400.it
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.it1 | 428400it6 | \([0, 0, 0, -258264075, -1594414079750]\) | \(40832710302042509761/91556816413125\) | \(4271674826570760000000000\) | \([2]\) | \(100663296\) | \(3.6084\) | |
428400.it2 | 428400it4 | \([0, 0, 0, -22014075, -5160329750]\) | \(25288177725059761/14387797265625\) | \(671277069225000000000000\) | \([2, 2]\) | \(50331648\) | \(3.2618\) | |
428400.it3 | 428400it2 | \([0, 0, 0, -14076075, 20233332250]\) | \(6610905152742241/35128130625\) | \(1638938062440000000000\) | \([2, 2]\) | \(25165824\) | \(2.9152\) | |
428400.it4 | 428400it1 | \([0, 0, 0, -14058075, 20287890250]\) | \(6585576176607121/187425\) | \(8744500800000000\) | \([2]\) | \(12582912\) | \(2.5687\) | \(\Gamma_0(N)\)-optimal* |
428400.it5 | 428400it3 | \([0, 0, 0, -6426075, 42135282250]\) | \(-629004249876241/16074715228425\) | \(-749981913697396800000000\) | \([2]\) | \(50331648\) | \(3.2618\) | |
428400.it6 | 428400it5 | \([0, 0, 0, 87227925, -41100947750]\) | \(1573196002879828319/926055908203125\) | \(-43206064453125000000000000\) | \([2]\) | \(100663296\) | \(3.6084\) |
Rank
sage: E.rank()
The elliptic curves in class 428400.it have rank \(1\).
Complex multiplication
The elliptic curves in class 428400.it do not have complex multiplication.Modular form 428400.2.a.it
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.