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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 25215h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25215.f7 | 25215h1 | \([1, 0, 0, -35, 12840]\) | \(-1/15\) | \(-71251563615\) | \([2]\) | \(16640\) | \(0.76136\) | \(\Gamma_0(N)\)-optimal |
25215.f6 | 25215h2 | \([1, 0, 0, -8440, 293567]\) | \(13997521/225\) | \(1068773454225\) | \([2, 2]\) | \(33280\) | \(1.1079\) | |
25215.f5 | 25215h3 | \([1, 0, 0, -16845, -390600]\) | \(111284641/50625\) | \(240474027200625\) | \([2, 2]\) | \(66560\) | \(1.4545\) | |
25215.f4 | 25215h4 | \([1, 0, 0, -134515, 18977882]\) | \(56667352321/15\) | \(71251563615\) | \([2]\) | \(66560\) | \(1.4545\) | |
25215.f8 | 25215h5 | \([1, 0, 0, 58800, -2917143]\) | \(4733169839/3515625\) | \(-16699585222265625\) | \([2]\) | \(133120\) | \(1.8011\) | |
25215.f2 | 25215h6 | \([1, 0, 0, -226970, -41617125]\) | \(272223782641/164025\) | \(779135848130025\) | \([2, 2]\) | \(133120\) | \(1.8011\) | |
25215.f3 | 25215h7 | \([1, 0, 0, -184945, -57494170]\) | \(-147281603041/215233605\) | \(-1022382059916218805\) | \([2]\) | \(266240\) | \(2.1477\) | |
25215.f1 | 25215h8 | \([1, 0, 0, -3630995, -2663397180]\) | \(1114544804970241/405\) | \(1923792217605\) | \([2]\) | \(266240\) | \(2.1477\) |
Rank
sage: E.rank()
The elliptic curves in class 25215h have rank \(1\).
Complex multiplication
The elliptic curves in class 25215h do not have complex multiplication.Modular form 25215.2.a.h
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.