Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+166x-9204\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+166xz^2-9204z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+215757x-430057458\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 370 \) | = | $2 \cdot 5 \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-37000000000 $ | = | $-1 \cdot 2^{9} \cdot 5^{9} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{510273943271}{37000000000} \) | = | $2^{-9} \cdot 5^{-9} \cdot 37^{-1} \cdot 61^{3} \cdot 131^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.70739920720767544399274644934\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.70739920720767544399274644934\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.996872612230027\dots$ | |||
Szpiro ratio: | $5.374291792619628\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.55062075505613942215128488880\dots$ | comment: Real Period
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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Tamagawa product: | $ 1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 0.55062075505613942215128488880 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 0.550620755 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.550621 \cdot 1.000000 \cdot 1}{1^2} \approx 0.550620755$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 324 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
$5$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
$37$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
---|---|---|
$3$ | 3B.1.2 | 9.24.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13320 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 37 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10 & 9 \\ 10647 & 13312 \end{array}\right),\left(\begin{array}{rr} 13303 & 18 \\ 13302 & 19 \end{array}\right),\left(\begin{array}{rr} 6670 & 9 \\ 3321 & 13312 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 3331 & 6678 \\ 0 & 371 \end{array}\right),\left(\begin{array}{rr} 10096 & 9 \\ 1935 & 76 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 6651 & 13312 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[13320])$ is a degree-$36273255874560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13320\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | nonsplit multiplicative | $4$ | \( 185 = 5 \cdot 37 \) |
$3$ | good | $2$ | \( 37 \) |
$5$ | nonsplit multiplicative | $6$ | \( 74 = 2 \cdot 37 \) |
$37$ | split multiplicative | $38$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 370c
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-136900.2-c4 |
$3$ | 3.1.1480.1 | \(\Z/2\Z\) | not in database |
$3$ | 3.1.4107.1 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.3241792000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.0.50602347.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$6$ | 6.0.36889110963.1 | \(\Z/9\Z\) | not in database |
$6$ | 6.0.26946027.1 | \(\Z/9\Z\) | not in database |
$6$ | 6.0.59140800.1 | \(\Z/6\Z\) | not in database |
$9$ | 9.1.164042283685824000.1 | \(\Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.50198942259523899975028947826347.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
$18$ | 18.0.726566512595229689293941092352000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.0.281486753600668545293576722736206000128000000.1 | \(\Z/18\Z\) | not in database |
$18$ | 18.0.150193475160708469172913491816448000000.1 | \(\Z/18\Z\) | not in database |
$18$ | 18.0.63722574141685329786964907655168000000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 37 |
---|---|---|---|---|
Reduction type | nonsplit | ord | nonsplit | split |
$\lambda$-invariant(s) | 10 | 0 | 0 | 1 |
$\mu$-invariant(s) | 0 | 2 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.