Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 53312.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.m1 | 53312cg2 | \([0, 1, 0, -9473, 336895]\) | \(6097250/289\) | \(4456521531392\) | \([2]\) | \(98304\) | \(1.1877\) | |
53312.m2 | 53312cg1 | \([0, 1, 0, -1633, -19041]\) | \(62500/17\) | \(131074162688\) | \([2]\) | \(49152\) | \(0.84111\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53312.m have rank \(2\).
Complex multiplication
The elliptic curves in class 53312.m do not have complex multiplication.Modular form 53312.2.a.m
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.