Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-258731208280x-50654853221670100\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-258731208280xz^2-50654853221670100z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-335315645930907x-2363351825963302392906\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1175681/4, 1175681/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 110670 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $2359330710565906457438184052500$ | = | $2^{2} \cdot 3 \cdot 5^{4} \cdot 7 \cdot 17^{16} \cdot 31^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{1915447099311696788795300773853656437121}{2359330710565906457438184052500} \) | = | $2^{-2} \cdot 3^{-1} \cdot 5^{-4} \cdot 7^{-1} \cdot 17^{-16} \cdot 31^{-4} \cdot 12419097997441^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $5.1080513193284656759127979465$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $5.1080513193284656759127979465$ |
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| $abc$ quality: | $Q$ | ≈ | $1.039784151271963$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.787874152908431$ | |||
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: | $\Omega$ | ≈ | $0.0066940258112350613250339743241$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2\cdot1\cdot2^{2}\cdot1\cdot2^{4}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.8546824307047027968347897079 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 6.854682431 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.006694 \cdot 1.000000 \cdot 256}{2^2} \\ & \approx 6.854682431\end{aligned}$$
Modular invariants
Modular form 110670.2.a.cm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 788529152 |
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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 semistable. There are 6 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 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $31$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 | 16.96.0.149 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 354144 = 2^{5} \cdot 3 \cdot 7 \cdot 17 \cdot 31 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 132805 & 32 \\ 132820 & 513 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 287743 & 32 \\ 199206 & 1 \end{array}\right),\left(\begin{array}{rr} 354113 & 32 \\ 354112 & 33 \end{array}\right),\left(\begin{array}{rr} 252992 & 29 \\ 205195 & 2562 \end{array}\right),\left(\begin{array}{rr} 30 & 31 \\ 235945 & 353988 \end{array}\right),\left(\begin{array}{rr} 208321 & 32 \\ 145840 & 513 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 351582 & 352139 \end{array}\right),\left(\begin{array}{rr} 342743 & 26 \\ 180246 & 351275 \end{array}\right)$.
The torsion field $K:=\Q(E[354144])$ is a degree-$3465112183426252800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/354144\Z)$.
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 | $4$ | \( 21 = 3 \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 36890 = 2 \cdot 5 \cdot 7 \cdot 17 \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 22134 = 2 \cdot 3 \cdot 7 \cdot 17 \cdot 31 \) |
| $7$ | split multiplicative | $8$ | \( 15810 = 2 \cdot 3 \cdot 5 \cdot 17 \cdot 31 \) |
| $17$ | split multiplicative | $18$ | \( 6510 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 110670.cm
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.592704.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.592704.1 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{42})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.5620768505856.46 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.351298031616.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.12745506816.8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.37201361225383936.107 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.1255023015100416.66 | \(\Z/16\Z\) | 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 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 7 | 17 | 31 |
|---|---|---|---|---|---|---|
| Reduction type | split | split | split | split | split | nonsplit |
| $\lambda$-invariant(s) | 14 | 1 | 1 | 5 | 1 | 0 |
| $\mu$-invariant(s) | 3 | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.