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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 319725.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 319725.t1 | 319725t3 | \([1, -1, 1, -2255979830, 41243628949422]\) | \(947531277805646290177/38367\) | \(51415449443859375\) | \([2]\) | \(75497472\) | \(3.6176\) | |
| 319725.t2 | 319725t6 | \([1, -1, 1, -468220955, -3169974639828]\) | \(8471112631466271697/1662662681263647\) | \(2228127010991287210988859375\) | \([2]\) | \(150994944\) | \(3.9642\) | |
| 319725.t3 | 319725t4 | \([1, -1, 1, -143700080, 618482054922]\) | \(244883173420511137/18418027974129\) | \(24681918997039573181390625\) | \([2, 2]\) | \(75497472\) | \(3.6176\) | |
| 319725.t4 | 319725t2 | \([1, -1, 1, -140998955, 644456072922]\) | \(231331938231569617/1472026689\) | \(1972656548812552640625\) | \([2, 2]\) | \(37748736\) | \(3.2711\) | |
| 319725.t5 | 319725t1 | \([1, -1, 1, -8643830, 10475024172]\) | \(-53297461115137/4513839183\) | \(-6048976211620611609375\) | \([2]\) | \(18874368\) | \(2.9245\) | \(\Gamma_0(N)\)-optimal |
| 319725.t6 | 319725t5 | \([1, -1, 1, 137602795, 2744569184172]\) | \(215015459663151503/2552757445339983\) | \(-3420939124073122939311609375\) | \([2]\) | \(150994944\) | \(3.9642\) |
Rank
sage: E.rank()
The elliptic curves in class 319725.t have rank \(1\).
Complex multiplication
The elliptic curves in class 319725.t do not have complex multiplication.Modular form 319725.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
