Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-31233x+2134962\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-31233xz^2+2134962z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2529900x+1548797625\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(77, 425)$ | $0.58019263732442557150673618224$ | $\infty$ |
| $(101, 23)$ | $2.4691944885935881482892466183$ | $\infty$ |
| $(102, 0)$ | $0$ | $2$ |
Integral points
\((-154,\pm 1808)\), \((-123,\pm 2025)\), \((-42,\pm 1836)\), \((53,\pm 791)\), \((77,\pm 425)\), \((101,\pm 23)\), \( \left(102, 0\right) \), \((103,\pm 9)\), \((111,\pm 153)\), \((327,\pm 5175)\)
Invariants
| Conductor: | $N$ | = | \( 66300 \) | = | $2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $76079250000$ | = | $2^{4} \cdot 3^{4} \cdot 5^{6} \cdot 13 \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{13478411517952}{304317} \) | = | $2^{14} \cdot 3^{-4} \cdot 13^{-1} \cdot 17^{-2} \cdot 937^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2020431615263115643279841769$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.16627514512261294055519380313$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9632064370361191$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.842689357082665$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3927946625170937303987377852$ |
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| Real period: | $\Omega$ | ≈ | $1.0063128264335647488031642284$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.4095228008749571987982854012 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.409522801 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.006313 \cdot 1.392795 \cdot 24}{2^2} \\ & \approx 8.409522801\end{aligned}$$
Modular invariants
Modular form 66300.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 122880 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $2$ | $I_{2}$ | split 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 | 4.6.0.3 |
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 $48$, genus $0$, and generators
$\left(\begin{array}{rr} 17681 & 15920 \\ 12380 & 10641 \end{array}\right),\left(\begin{array}{rr} 5303 & 0 \\ 0 & 26519 \end{array}\right),\left(\begin{array}{rr} 26513 & 8 \\ 26512 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 25201 & 5310 \\ 10630 & 15931 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 28 & 75 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 13261 & 15920 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13111 & 5310 \\ 25890 & 21211 \end{array}\right),\left(\begin{array}{rr} 14286 & 5305 \\ 22055 & 15916 \end{array}\right)$.
The torsion field $K:=\Q(E[26520])$ is a degree-$1513657875824640$ 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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 22100 = 2^{2} \cdot 5^{2} \cdot 13 \cdot 17 \) |
| $5$ | additive | $14$ | \( 2652 = 2^{2} \cdot 3 \cdot 13 \cdot 17 \) |
| $13$ | split multiplicative | $14$ | \( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 66300l
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2652d1, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.20800.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.73116160000.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | ord | ord | split | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 4 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.