Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-35564025x-6328526875\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-35564025xz^2-6328526875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-46090977075x-294572385227250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6050, -3025)$ | $0$ | $2$ |
Integral points
\( \left(6050, -3025\right) \)
Invariants
| Conductor: | $N$ | = | \( 143650 \) | = | $2 \cdot 5^{2} \cdot 13^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $2861587305168728825000000$ | = | $2^{6} \cdot 5^{8} \cdot 13^{12} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{65959341605440921}{37942580187200} \) | = | $2^{-6} \cdot 5^{-2} \cdot 11^{3} \cdot 13^{-6} \cdot 17^{-3} \cdot 23^{3} \cdot 1597^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3823875805156110686618747342$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2951939455677925133347513468$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0055768958758147$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.370392688951174$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.067241776110684329533573693250$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot2\cdot2^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.2276052533128478176115372760 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.227605253 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.067242 \cdot 1.000000 \cdot 48}{2^2} \\ & \approx 3.227605253\end{aligned}$$
Modular invariants
Modular form 143650.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27869184 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $13$ | $4$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $17$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 26520 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 2039 & 15900 \\ 6930 & 15839 \end{array}\right),\left(\begin{array}{rr} 5303 & 0 \\ 0 & 26519 \end{array}\right),\left(\begin{array}{rr} 13261 & 10620 \\ 5310 & 10681 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 13491 & 3760 \\ 8150 & 19661 \end{array}\right),\left(\begin{array}{rr} 13426 & 15915 \\ 23685 & 5296 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 26509 & 12 \\ 26508 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17681 & 10620 \\ 4420 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 26470 & 26511 \end{array}\right)$.
The torsion field $K:=\Q(E[26520])$ is a degree-$756828937912320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/26520\Z)$.
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 | $4$ | \( 71825 = 5^{2} \cdot 13^{2} \cdot 17 \) |
| $3$ | good | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 5746 = 2 \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $98$ | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 143650.v
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2210.a2, its twist by $65$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-195}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-195})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.4.4596800.2 | \(\Z/4\Z\) | not in database |
| $6$ | 6.2.5005040625.6 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.6106734799360000.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | 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 |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.1137054347136195886516234353900185156604484995000000000000.1 | \(\Z/18\Z\) | 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 | 13 | 17 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | add | add | split |
| $\lambda$-invariant(s) | 4 | 0 | - | - | 1 |
| $\mu$-invariant(s) | 0 | 1 | - | - | 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 $0$.