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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 327990.bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
327990.bn1 | 327990bn4 | \([1, 0, 0, -7024470, -6561320400]\) | \(64443098670429961/6032611833300\) | \(3588338204987404389300\) | \([2]\) | \(41287680\) | \(2.8744\) | |
327990.bn2 | 327990bn2 | \([1, 0, 0, -1557970, 633686900]\) | \(703093388853961/115124490000\) | \(68478731470231290000\) | \([2, 2]\) | \(20643840\) | \(2.5278\) | |
327990.bn3 | 327990bn1 | \([1, 0, 0, -1490690, 700388292]\) | \(615882348586441/21715200\) | \(12916707380179200\) | \([4]\) | \(10321920\) | \(2.1812\) | \(\Gamma_0(N)\)-optimal |
327990.bn4 | 327990bn3 | \([1, 0, 0, 2832050, 3560074232]\) | \(4223169036960119/11647532812500\) | \(-6928224148987720312500\) | \([2]\) | \(41287680\) | \(2.8744\) |
Rank
sage: E.rank()
The elliptic curves in class 327990.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 327990.bn do not have complex multiplication.Modular form 327990.2.a.bn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.