Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-5556819x-5243639758\)
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(homogenize, simplify) |
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\(y^2z=x^3-5556819xz^2-5243639758z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5556819x-5243639758\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3801709, 7412560020)$ | $5.5199244753578158947484697147$ | $\infty$ |
| $(2734, 0)$ | $0$ | $2$ |
Integral points
\( \left(2734, 0\right) \), \((3801709,\pm 7412560020)\)
Invariants
| Conductor: | $N$ | = | \( 101232 \) | = | $2^{4} \cdot 3^{2} \cdot 19 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-896741736952500099072$ | = | $-1 \cdot 2^{11} \cdot 3^{8} \cdot 19 \cdot 37^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{12709983426958940834}{600633986620491} \) | = | $-1 \cdot 2 \cdot 3^{-2} \cdot 19^{-1} \cdot 37^{-8} \cdot 1852273^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7832546122204166415693102102$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5985635523730785955725581471$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9754441228809531$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.0570801106326115$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.5199244753578158947484697147$ |
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| Real period: | $\Omega$ | ≈ | $0.049030560201517901402153843914$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2^{3} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.6606396574996323158051410621 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.660639657 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.049031 \cdot 5.519924 \cdot 128}{2^2} \\ & \approx 8.660639657\end{aligned}$$
Modular invariants
Modular form 101232.2.a.ba
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3407872 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $19$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $37$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
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 | 8.24.0.92 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 33744 = 2^{4} \cdot 3 \cdot 19 \cdot 37 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 33646 & 33731 \end{array}\right),\left(\begin{array}{rr} 10672 & 11253 \\ 24819 & 33730 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 30940 & 19689 \\ 16875 & 5626 \end{array}\right),\left(\begin{array}{rr} 15505 & 22512 \\ 312 & 11377 \end{array}\right),\left(\begin{array}{rr} 33729 & 16 \\ 33728 & 17 \end{array}\right),\left(\begin{array}{rr} 12653 & 11232 \\ 26706 & 13931 \end{array}\right),\left(\begin{array}{rr} 22495 & 0 \\ 0 & 33743 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 33740 & 33741 \end{array}\right)$.
The torsion field $K:=\Q(E[33744])$ is a degree-$1378383723233280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/33744\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 171 = 3^{2} \cdot 19 \) |
| $3$ | additive | $8$ | \( 11248 = 2^{4} \cdot 19 \cdot 37 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 101232.ba
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 16872.c3, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-38}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{114}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-38})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{38})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.691798081536.9 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ss | ord | ord | ord | nonsplit | ss | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1,3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 2 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 0 | 0,0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.