Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-102x+385\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-102xz^2+385z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-131571x+18368910\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Torsion generators
\( \left(5, 0\right) \)
Integral points
\( \left(5, 0\right) \), \( \left(5, -6\right) \)
Invariants
Conductor: | \( 57 \) | = | $3 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1539 $ | = | $3^{4} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{115714886617}{1539} \) | = | $3^{-4} \cdot 11^{3} \cdot 19^{-1} \cdot 443^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $-0.24500866220282609366943236987\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.24500866220282609366943236987\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9811108308373943\dots$ | |||
Szpiro ratio: | $6.300784590885129\dots$ |
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: | $4.3412090890706928922224104326\dots$ | 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: | $ 4 $ = $ 2^{2}\cdot1 $ | 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: | $4$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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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) $ ≈ $ 1.0853022722676732230556026082 $ | 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
$\displaystyle 1.085302272 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.341209 \cdot 1.000000 \cdot 4}{4^2} \approx 1.085302272$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 6 | 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 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
$19$ | $1$ | $I_{1}$ | nonsplit 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 |
---|---|---|
$2$ | 2B | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 449 & 8 \\ 448 & 9 \end{array}\right),\left(\begin{array}{rr} 388 & 1 \\ 407 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 450 & 451 \end{array}\right),\left(\begin{array}{rr} 288 & 65 \\ 313 & 360 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 308 & 33 \end{array}\right),\left(\begin{array}{rr} 403 & 402 \\ 298 & 67 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$189112320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 19 \) |
$3$ | split multiplicative | $4$ | \( 19 \) |
$19$ | nonsplit multiplicative | $20$ | \( 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 57.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.76.1-171.1-i5 |
$4$ | 4.0.2736.1 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.3083198857216.5 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.43237380096.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.8.249739107434496.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.23085974187.2 | \(\Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 19 |
---|---|---|---|
Reduction type | ord | split | nonsplit |
$\lambda$-invariant(s) | 0 | 3 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.