Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-3773654267x+93240113640041\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-3773654267xz^2+93240113640041z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-60378468267x+5967306894494374\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{285141}{4}, \frac{285137}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 106470 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-316340292848806015646929458750 $ | = | $-1 \cdot 2 \cdot 3^{8} \cdot 5^{4} \cdot 7^{3} \cdot 13^{18} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1688971789881664420008241}{89901485966373558750} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{-3} \cdot 13^{-12} \cdot 47^{3} \cdot 157^{3} \cdot 16139^{3}$ | 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: | $4.4195625737916419091108126655\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.5877817507268186953864463263\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0262585206734374\dots$ | |||
Szpiro ratio: | $6.725819736397446\dots$ |
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: | $0.030193273349725956294461763863\dots$ | 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: | $ 64 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot1\cdot2^{2} $ | 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: | $9$ = $3^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 4.3478313623605377064024939962 $ | 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
$\displaystyle 4.347831362 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.030193 \cdot 1.000000 \cdot 64}{2^2} \approx 4.347831362$
Modular invariants
Modular form 106470.2.a.fd
For more coefficients, see the Downloads section to the right.
Modular degree: | 148635648 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $4$ | $I_{2}^{*}$ | Additive | -1 | 2 | 8 | 2 |
$5$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$7$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$13$ | $4$ | $I_{12}^{*}$ | Additive | 1 | 2 | 18 | 12 |
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 | 4.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 9099 & 7276 \\ 10900 & 5379 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 9614 & 1691 \end{array}\right),\left(\begin{array}{rr} 5922 & 7759 \\ 3677 & 6692 \end{array}\right),\left(\begin{array}{rr} 8191 & 24 \\ 8190 & 1 \end{array}\right),\left(\begin{array}{rr} 10897 & 24 \\ 10896 & 25 \end{array}\right),\left(\begin{array}{rr} 6719 & 10896 \\ 4188 & 10631 \end{array}\right),\left(\begin{array}{rr} 4696 & 21 \\ 4395 & 10546 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 8737 & 24 \\ 6564 & 289 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$4869303828480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 106470.fd
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2730.bd3, its twist by $-39$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-546}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{13}) \) | \(\Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-14}, \sqrt{39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{13}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{3}, \sqrt{13})\) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{13}, \sqrt{-42})\) | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.3891919590000.8 | \(\Z/6\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.290198039040000.12 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.364024420171776.120 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.6.14586947045846472881615931502076904200889300000000.4 | \(\Z/18\Z\) | 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 | 7 | 13 |
---|---|---|---|---|---|
Reduction type | split | add | split | nonsplit | add |
$\lambda$-invariant(s) | 5 | - | 1 | 0 | - |
$\mu$-invariant(s) | 1 | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.