Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-27x+45\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-27xz^2+45z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-35019x+2204550\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Torsion generators
\( \left(-6, 3\right) \), \( \left(2, -1\right) \)
Integral points
\( \left(-6, 3\right) \), \( \left(2, -1\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 546 \) | = | $2 \cdot 3 \cdot 7 \cdot 13$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $298116 $ | = | $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{2181825073}{298116} \) | = | $2^{-2} \cdot 3^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 1297^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $-0.22260093612944015531022418482\dots$ | ||
Stable Faltings height: | $-0.22260093612944015531022418482\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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Real period: | $2.9547312603904991530188126999\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | $ 16 $ = $ 2\cdot2\cdot2\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $4$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Special value: | $ L(E,1) $ ≈ $ 2.9547312603904991530188126999 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 96 | ||
$ \Gamma_0(N) $-optimal: | no | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
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$2$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$13$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2Cs | 8.12.0.1 |
The image of the adelic Galois representation has level $2184$, index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 2017 & 4 \\ 1850 & 9 \end{array}\right),\left(\begin{array}{rr} 1563 & 2 \\ 1870 & 2183 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1457 & 4 \\ 730 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1093 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1639 & 4 \\ 0 & 1 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
$p$ | 2 | 3 | 7 | 13 |
---|---|---|---|---|
Reduction type | split | split | split | nonsplit |
$\lambda$-invariant(s) | 2 | 1 | 1 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 546.g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{7}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.194365577860272.8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | 16.0.132513778481397717569346994176.17 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.