Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-19440x+1048135\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-19440xz^2+1048135z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-311043x+66769598\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{323}{4}, -\frac{323}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 45 \) | = | $3^{2} \cdot 5$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $295245 $ | = | $3^{10} \cdot 5 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1114544804970241}{405} \) | = | $3^{-4} \cdot 5^{-1} \cdot 103681^{3}$ | 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: | $0.84017575281115636767125824276\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.29086960847710152197363562430\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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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: | $1.8431817532792632294955869222\dots$ | 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: | $ 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: | $2$ | 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) $ ≈ $ 0.92159087663963161474779346108 $ | 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
$\displaystyle 0.921590877 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.843182 \cdot 1.000000 \cdot 2}{2^2} \approx 0.921590877$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 32 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $1$ | $I_{1}$ | Non-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 |
---|---|---|
$2$ | 2B | 32.96.0.51 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 480 = 2^{5} \cdot 3 \cdot 5 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 449 & 32 \\ 448 & 33 \end{array}\right),\left(\begin{array}{rr} 31 & 32 \\ 390 & 1 \end{array}\right),\left(\begin{array}{rr} 112 & 25 \\ 23 & 306 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 318 & 395 \end{array}\right),\left(\begin{array}{rr} 421 & 32 \\ 436 & 33 \end{array}\right),\left(\begin{array}{rr} 137 & 454 \\ 138 & 469 \end{array}\right)$.
The torsion field $K:=\Q(E[480])$ is a degree-$11796480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/480\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 45a
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a5, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.5.1-405.1-a9 |
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/4\Z\) | 2.2.12.1-75.1-a8 |
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | 2.2.60.1-15.1-b9 |
$4$ | \(\Q(\sqrt{3}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{10})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{3})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.324000000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.82944000000.23 | \(\Z/8\Z\) | Not in database |
$8$ | 8.8.3317760000.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.8.21233664000000.4 | \(\Z/16\Z\) | Not in database |
$8$ | 8.8.21233664000000.1 | \(\Z/16\Z\) | Not in database |
$8$ | 8.2.110716875.1 | \(\Z/6\Z\) | Not in database |
$16$ | 16.0.26873856000000000000.6 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/16\Z\) | Not in database |
$16$ | 16.16.450868486864896000000000000.1 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 5 |
---|---|---|---|
Reduction type | ord | add | nonsplit |
$\lambda$-invariant(s) | 1 | - | 0 |
$\mu$-invariant(s) | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.