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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2310.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2310.h1 | 2310g7 | \([1, 0, 1, -25467989, -49471735264]\) | \(1826870018430810435423307849/7641104625000000000\) | \(7641104625000000000\) | \([2]\) | \(165888\) | \(2.8322\) | |
2310.h2 | 2310g6 | \([1, 0, 1, -1616469, -747850208]\) | \(467116778179943012100169/28800309694464000000\) | \(28800309694464000000\) | \([2, 2]\) | \(82944\) | \(2.4857\) | |
2310.h3 | 2310g4 | \([1, 0, 1, -437774, -9825928]\) | \(9278380528613437145689/5328033205714065000\) | \(5328033205714065000\) | \([6]\) | \(55296\) | \(2.2829\) | |
2310.h4 | 2310g3 | \([1, 0, 1, -305749, 50640416]\) | \(3160944030998056790089/720291785342976000\) | \(720291785342976000\) | \([2]\) | \(41472\) | \(2.1391\) | |
2310.h5 | 2310g2 | \([1, 0, 1, -286854, 58872856]\) | \(2610383204210122997209/12104550027662400\) | \(12104550027662400\) | \([2, 6]\) | \(27648\) | \(1.9364\) | |
2310.h6 | 2310g1 | \([1, 0, 1, -286534, 59011352]\) | \(2601656892010848045529/56330588160\) | \(56330588160\) | \([6]\) | \(13824\) | \(1.5898\) | \(\Gamma_0(N)\)-optimal |
2310.h7 | 2310g5 | \([1, 0, 1, -141054, 118709176]\) | \(-310366976336070130009/5909282337130963560\) | \(-5909282337130963560\) | \([6]\) | \(55296\) | \(2.2829\) | |
2310.h8 | 2310g8 | \([1, 0, 1, 1263531, -3122122208]\) | \(223090928422700449019831/4340371122724101696000\) | \(-4340371122724101696000\) | \([2]\) | \(165888\) | \(2.8322\) |
Rank
sage: E.rank()
The elliptic curves in class 2310.h have rank \(1\).
Complex multiplication
The elliptic curves in class 2310.h do not have complex multiplication.Modular form 2310.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.