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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 351975.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 351975.q1 | 351975q7 | \([1, 0, 0, -1173250188, 15467898628617]\) | \(242970740812818720001/24375\) | \(17917864833984375\) | \([2]\) | \(63700992\) | \(3.4664\) | |
| 351975.q2 | 351975q5 | \([1, 0, 0, -73328313, 241680112992]\) | \(59319456301170001/594140625\) | \(436747955328369140625\) | \([2, 2]\) | \(31850496\) | \(3.1198\) | |
| 351975.q3 | 351975q8 | \([1, 0, 0, -71568438, 253832049867]\) | \(-55150149867714721/5950927734375\) | \(-4374478719234466552734375\) | \([2]\) | \(63700992\) | \(3.4664\) | |
| 351975.q4 | 351975q3 | \([1, 0, 0, -4693188, 3584864367]\) | \(15551989015681/1445900625\) | \(1062869824087119140625\) | \([2, 2]\) | \(15925248\) | \(2.7732\) | |
| 351975.q5 | 351975q2 | \([1, 0, 0, -1038063, -344395008]\) | \(168288035761/27720225\) | \(20376912603800390625\) | \([2, 2]\) | \(7962624\) | \(2.4266\) | |
| 351975.q6 | 351975q1 | \([1, 0, 0, -992938, -380901133]\) | \(147281603041/5265\) | \(3870258804140625\) | \([2]\) | \(3981312\) | \(2.0801\) | \(\Gamma_0(N)\)-optimal |
| 351975.q7 | 351975q4 | \([1, 0, 0, 1895062, -1937081883]\) | \(1023887723039/2798036865\) | \(-2056814209131297890625\) | \([2]\) | \(15925248\) | \(2.7732\) | |
| 351975.q8 | 351975q6 | \([1, 0, 0, 5459937, 16976836242]\) | \(24487529386319/183539412225\) | \(-134918333536676487890625\) | \([2]\) | \(31850496\) | \(3.1198\) |
Rank
sage: E.rank()
The elliptic curves in class 351975.q have rank \(0\).
Complex multiplication
The elliptic curves in class 351975.q do not have complex multiplication.Modular form 351975.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
