Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-9727138x-10940879608\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-9727138xz^2-10940879608z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12606370875x-510419859878250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14303/4, -14303/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 139650 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $7198733304367640437500$ | = | $2^{2} \cdot 3^{18} \cdot 5^{6} \cdot 7^{7} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{55369510069623625}{3916046302812} \) | = | $2^{-2} \cdot 3^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 19^{-2} \cdot 31^{3} \cdot 2459^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9412949780765291545659041449$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1636209473318223147128481066$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9950582420479482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.054902234974544$ | |||
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: | $\Omega$ | ≈ | $0.085868678208261041927994802139$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ 2\cdot( 2 \cdot 3^{2} )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.1825448309947950188156257540 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 6.182544831 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.085869 \cdot 1.000000 \cdot 288}{2^2} \\ & \approx 6.182544831\end{aligned}$$
Modular invariants
Modular form 139650.2.a.jq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15925248 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 23940 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 14363 & 0 \\ 0 & 23939 \end{array}\right),\left(\begin{array}{rr} 9586 & 4815 \\ 16015 & 17146 \end{array}\right),\left(\begin{array}{rr} 21884 & 14355 \\ 10865 & 18974 \end{array}\right),\left(\begin{array}{rr} 10861 & 14400 \\ 15150 & 9841 \end{array}\right),\left(\begin{array}{rr} 23905 & 36 \\ 23904 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 5321 & 14400 \\ 5330 & 361 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[23940])$ is a degree-$51468809011200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/23940\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $32$ | \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 139650cq
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798e2, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.2274300.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.118276472538000.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.273788130875.2 | \(\Z/18\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.4055193344160000.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.253449584010000.25 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
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 | 19 |
|---|---|---|---|---|---|
| Reduction type | split | split | add | add | nonsplit |
| $\lambda$-invariant(s) | 6 | 5 | - | - | 0 |
| $\mu$-invariant(s) | 0 | 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 $0$.